Handbook of Graph Theory

Published on January 2017 | Categories: Documents | Downloads: 52 | Comments: 0 | Views: 478
of 1155
Download PDF   Embed   Report

Comments

Content

DISCRETE MATHEMATICS AND ITS APPLICATIONS
Series Editor KENNETH H. ROSEN

HANDBOOK OF

GRAPH THEORY
EDITED BY

JONATHAN L. GROSS
JAY YELLEN

CRC PR E S S
Boca Raton London New York Washington, D.C.

DISCRETE
MATHEMATICS
and
ITS APPLICATIONS
Series Editor

Kenneth H. Rosen, Ph.D.
AT&T Laboratories
Middletown, New Jersey

Charles J. Colbourn and Jeffrey H. Dinitz, The CRC Handbook of Combinatorial Designs
Charalambos A. Charalambides, Enumerative Combinatorics
Steven Furino, Ying Miao, and Jianxing Yin, Frames and Resolvable Designs: Uses,
Constructions, and Existence
Randy Goldberg and Lance Riek, A Practical Handbook of Speech Coders
Jacob E. Goodman and Joseph O’Rourke, Handbook of Discrete and Computational Geometry
Jonathan L. Gross and Jay Yellen, Graph Theory and Its Applications
Jonathan L. Gross and Jay Yellen, Handbook of Graph Theory
Darrel R. Hankerson, Greg A. Harris, and Peter D. Johnson, Introduction to Information
Theory and Data Compression
Daryl D. Harms, Miroslav Kraetzl, Charles J. Colbourn, and John S. Devitt, Network Reliability:
Experiments with a Symbolic Algebra Environment
David M. Jackson and Terry I. Visentin, An Atlas of Smaller Maps in Orientable and
Nonorientable Surfaces
Richard E. Klima, Ernest Stitzinger, and Neil P. Sigmon, Abstract Algebra Applications
with Maple
Patrick Knupp and Kambiz Salari, Verification of Computer Codes in Computational Science
and Engineering
Donald L. Kreher and Douglas R. Stinson, Combinatorial Algorithms: Generation Enumeration
and Search
Charles C. Lindner and Christopher A. Rodgers, Design Theory
Alfred J. Menezes, Paul C. van Oorschot, and Scott A. Vanstone, Handbook of Applied
Cryptography
Richard A. Mollin, Algebraic Number Theory
Richard A. Mollin, Fundamental Number Theory with Applications
Richard A. Mollin, An Introduction to Crytography
Richard A. Mollin, Quadratics

Continued Titles
Richard A. Mollin, RSA and Public-Key Cryptography
Kenneth H. Rosen, Handbook of Discrete and Combinatorial Mathematics
Douglas R. Shier and K.T. Wallenius, Applied Mathematical Modeling: A Multidisciplinary
Approach
Douglas R. Stinson, Cryptography: Theory and Practice, Second Edition
Roberto Togneri and Christopher J. deSilva, Fundamentals of Information Theory and
Coding Design
Lawrence C. Washington, Elliptic Curves: Number Theory and Cryptography

8522 disclaimer.fm Page 1 Tuesday, November 4, 2003 12:31 PM

Library of Congress Cataloging-in-Publication Data
Handbook of graph theory / editors-in-chief, Jonathan L. Gross, Jay Yellen.
p. cm. — (Discrete mathematics and its applications)
Includes bibliographical references and index.
ISBN 1-58488-090-2 (alk. paper)
1. Graph theory—Handbooks, manuals, etc. I. Gross, Jonathan L. II. Yellen, Jay.
QA166.H36 2003
511'.5—dc22

2003065270

This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with
permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish
reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials
or for the consequences of their use.
Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical,
including photocopying, microÞlming, and recording, or by any information storage or retrieval system, without prior
permission in writing from the publisher.
All rights reserved. Authorization to photocopy items for internal or personal use, or the personal or internal use of speciÞc
clients, may be granted by CRC Press LLC, provided that $1.50 per page photocopied is paid directly to Copyright Clearance
Center, 222 Rosewood Drive, Danvers, MA 01923 USA. The fee code for users of the Transactional Reporting Service is
ISBN 1-58488-090-2/04/$0.00+$1.50. The fee is subject to change without notice. For organizations that have been granted
a photocopy license by the CCC, a separate system of payment has been arranged.
The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works,
or for resale. SpeciÞc permission must be obtained in writing from CRC Press LLC for such copying.
Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for
identiÞcation and explanation, without intent to infringe.

Visit the CRC Press Web site at www.crcpress.com

No claim to original U.S. Government works
International Standard Book Number 1-58488-090-2
Library of Congress Card Number 2003065270
Printed in the United States of America 1 2 3 4 5 6 7 8 9 0
Printed on acid-free paper

PREFACE
 
    
         

 
               



 
!  
        
    

 

   

    
"

   
  
   
 
   
  
 
 "      
 


 



     #         
  


  
$ 

   
   

 %
& 
 
Format

'     (  
  

    
   )

             

      
 $ * ( 
    )
 +
 (     
  

  


"  
  

   $  ,-  +  
 
 . 
 
 /   
  
   

!   
  ) 
  
  (     
.   


 0        

   
 
     '      

  



 )      
   
  

'        
 
        

  !      

    
  1   
  (  
 
       "  
  

      
 
 
Terminology and Notations

2
              

 
  
 3   4 

    
  
  

 .     
 
  )
 

    
     
   (    
 
    
      
       

      

 5      
 

    
  
 
    0     ( 


     
   
     6 
             

         
 ' 
      


    
    7  
  
   



'    
       )   



8     
   
   
/    
  
     
  
   



       


    
  7 


 "  
  
 

  
  )
  
     )
  
* 

     


  ' 
 
  2(


         
  

     


  
9   
        ( )
   
   
  
 


:  Æ
        6  
 
   

     7   * 
   ( ;

6         )
     
       2   )
.
Feedback

" 
   
  (    
   
 

 
(   
  
 

 

Acknowledgements

+ 

(   ( 5    <        
     
    ( 8
    =   
1
<. >

      



    !

  + 


(   (  =    
   
 

  
  

   
     
   
.
?   2  ?  ;



?      (  0  ? ?    0
 

?     (  5  "  

About the Editors
?   2  <   
   

 : 8    

 
  

 
       0
 < 
  /



  '5= <
/


      
    Æ  9 
   9 

 /      

  /  
<  2      
      


     5

@     '5=
" 
   
 
   

    

  

   
  
    

 8    
    
     
 : 
    2  "  0 
    
 2   8

    
 A 9(
    (   9 
"

:
8
  ( 
          ? 
;

        "  + " ( 0
  (             (
 
     


 
<  
 : <  2     =   >
 
< : 8    (    ='"     <>
  B 


  >   


?  ;

  0  <   =  

 

 

 8   5  =  = 
   <
 :  9 ;(   

(  ! 
  
    :
 > ;

     
 


 
0

 

   :  9 ;(  / 
 /
 '   "
       

     ,   8     


  1 : 2 '  
"
  
 :
'    (      
   ?   @ 2       
   '5=  
         <
  
     
     

 ! "   
   
8
       
  (
   



    
          9 

 /    9 ;(   ' 0

C    6 
<     

 


> ;

  

   
       ! 
 


  
   
 8      


   
$   

        
 


CONTRIBUTORS
0
 A 0


 / 

5  0



' /

>  0 

= 
0
/


@

+ 5(

@ /



 #  

#   !  $   
#   ! % 
&  #    ' ( 

5D
5

D 

#   ! 
  $ 
 $ #)

#   ! 
#   !  - %1$ 
  #   !  $ /
5
   #  

8 /


  #   ! % $  

 5 5

0
 /$

  5 

8 
 9 2 

1  5 (

= ( 2


2    


 ? 2


?  

?   @ 2

#   ! 

#   !   ' 
#   ! * $ #)
(   #  
+ , #  

>

#  !  -. / 01$ 

   #  

#   !  
  &   
3 #  

 #  

2 2 


$ #   ! .  $ #)

0 E >

" + 8 

>*   +

= 
0 8

#   ! ($   
32 & 4 '2 2
 .  $ /0

= 
>

#   !  
$   

?
/F 

  #   !  $ /

3     / #  
#   ! *$ / !

2 

 / '
 

#   !  - %1$ 

? E

#   ! / $  

? @ 

#   ! $ 


 


 #  

2 

 @

<
E (


 5 = 

 "  

/  ==

E   %E"& "
  

#   ! &$ 
/ 
    !  

< (  =  

#   ! &


 ! (  

5 #  
#   ! 9

 0 "

   !  

 9 

0
 " (

A
 9( 

"  + " (

?  )


H
 " $

6 *  #  $ 7
#   ! 
.   / #  

 2  < (

   !  

" G$ < (

#   ! .8
8 $ / 

= 
<


% 
 #  

<G$ <G(

#   ! .8
8 $ / 

E 5 

!  / #   / 

> 
  

  #  

>   @

     !  

/ #   ! * 6 /  5
 #  


    ! / 
#   ! % 02$
 

0 A

#   ! ' 

= ( 1 + (

/  #  

0  " +

(   #  

 ? +


 9 #  $ #)

9
 +


#   ! ($   

?  ;



 

< H 

(   #  

CONTENTS
Preface
1.

INTRODUCTION to GRAPHS
 

  
    

 

 

  
   
  
  
   




1



2.

       

    
   !    " #

     
  &  &   " #

 
  
' & (
   &  
             !






56

$%




DIRECTED GRAPHS


) &  * 
 "  
 

 &  +&&
& 
 "#   
  
 $  



4.



GRAPH REPRESENTATION


3.



126

'
$
%

CONNECTIVITY and TRAVERSABILITY
'
'
'
''
'
'$
'

 & ( , "    - & 
 
# %&     %
.
 
 '   %
(
  "  " #

    
)/   -0&
  $ ) 

 
   
(
 -
  " #

   
   &   & ( 
 
# %&     %



193
'
 '


$
 



5.

COLORINGS and RELATED TOPICS



'

$

6.

'
$
%
'
'
''
' 

ALGEBRAIC GRAPH THEORY
$
$
$
$'
$
$$

7.

 

 *
 +
   &   

 *
 +
! -  
0
   
&   & 1  
 ( ,,
"
& 
  (
+
&    #

 - ,   ),.     /  $




340

+   
   - 


 
   
#(
.  
  $ "(, 
  2&  -&
 $ 
  0$ 1 
 ,
-& 
  
 ( )
*  
*      
 ,
23



484
'%

 $


 '


TOPOLOGICAL GRAPH THEORY



'

$

%


  -
&
 ,4+ 

   ,4+ 4(
*  *3 !#
  
  #  
    


2
 
    


 
  
 ,
 (
*
   (
   ( 
 )  ( (

  
 "  5,
      
    



610
$
$
$'
$$
$%'
$$


$


8.

ANALYTIC GRAPH THEORY
%

.3 
  
 6 6
  , 5 !
%   
 5(
 ,
%     
 # % 
%'  " ##
 & *  
  % +



9.

%%
%
%
%$
%$%

GRAPHICAL MEASUREMENT


 &  
       *
      
  
  ' 
 (  '
 
& 
 %  ( 

' )4 
     ,



10.

787

872
% 
%%
 

'

GRAPHS in COMPUTER SCIENCE

952



-&

 '  5 
 &  +
  
%
 , ), 
( 7  %((  
## % 7
 4

 
 
##     ,


' +
    & (
   &  
'$
             !


$

11.

NETWORKS and FLOWS


*3 
4
 / "
 *  
4
 
 %
(
 * &  + 
 )
 "
'  &   5 4 6   * 

  
  (   )! ",(8!



1074
 
%




Chapter

1

INTRODUCTION TO GRAPHS
1.1

FUNDAMENTALS of GRAPH THEORY
   


 

 

 






1.2

FAMILIES of GRAPHS and DIGRAPHS


    
    

1.3

HISTORY of GRAPH THEORY
  
  !" 
 #
GLOSSARY

2

Chapter 1

1.1

INTRODUCTION TO GRAPHS

FUNDAMENTALS OF GRAPH THEORY
   


 

 

 





   

 

  

     
  


Introduction
                  
      !"       "  "    
          #       
"      " #"    $     
 

1.1.1 Graphs and Digraphs
%   #&        #    #
   " '        "      " (  
                  $   
         #       #   
Basic Terminology
DEFINITIONS

)

% 

* +  ,      







    
+

,
      



¯   
¯

¯ -                "

  
 %           

    

   #       +  + ,   + ,,    
./   /           



) ' .
   

)
)

% .

         "      #  
   "   
  
    .     &  #   

   &    #    


Section 1.1

)
)
)
)

3

Fundamentals of Graph Theory


   

            
%   
      &       
% 
                  

%
 
   #    
#  

   .   

)  
  #       
#  

     #   




)

%
       &          

  
 %     0/1  01    #  
. 

  01      

  
 '   "   0 1       
    "     /  
  

 

EXAMPLE

)

%        * +  ,     2     '  ./
*       /  *  
   
    
    /  
     "      /



Figure 1.1.1




REMARKS

)

%    3      /      "   
 "            "      
                  4 "  
 $    "     



) ( "    "  %            /
        56 7 +.    /"     6
     ,
Simple Graphs

8        
"
      #
 #       #    # #  

DEFINITIONS

)
)

%
        /   / 
%           .    

4

)

Chapter 1

%     

INTRODUCTION TO GRAPHS

 ./   /  

Edge Notation for Simple Adjacencies and for Multi-edges

  %   &            #  #  &./
            ( "      + ,   
    "      #      /  . "  
   "         #      "   2 
  #
EXAMPLE

)

       2   
  /

Figure 1.1.2



*    




General Graphs

8    9  /      8"  / 
          "     
    +   / ,   +  :,

  
 %    0 1   /   / 
   "  "   "        
DEFINITIONS



) %  

       / +'     / ,
           

)
)


      
      

     

  . 





  &   

/

EXAMPLES

)

    2       #3   4 

Figure 1.1.3

     


Section 1.1

)

       2   

Figure 1.1.4

)

5

Fundamentals of Graph Theory

  

 
 

 /     2   ;

Figure 1.1.5

 !
  " !#!" 

Attributes

%         # #      # 
                    
       #" #  #        "
    $ + "    ,     + "    ,
%   # +     ,  $
      
  ;
DEFINITIONS



) %         ./      #
 # 



) % 
         /      #
 # 
Digraphs

%   #                  %/
      !     &      $  <  
       #     #     /
       8"      "   
 $  "
    !           
 # #                
 #   #        .     
     
DEFINITIONS



) %
 

 +  ,     "         
  "            
   
 +,   +,"  

  
 %       #
 
     
 

   +         /    .,

6

Chapter 1

INTRODUCTION TO GRAPHS

  '      "         


)
)
)
)

%                    
%
  +
 
,     
%

    

     

  /    /

% 
 + 
 
,      #
      '   . "       0 1
              



)  
            
                   
 + "      /  ,
Ordered-Pair Representation of Arcs

  '    "    .   .    
+ , +   , =  /   #"       


   +' $ 
&  

, 2    

#&/     "    /        
    >   #&         /
 
&  
 ?            #    
                   !  
  
   /        !       
        $  + "      
  , =           #  "   /
       #    #   - "   
    #   "        #     
    "              
9    >   #&
EXAMPLES

)

      2   @          
           /)  
  
 

Figure 1.1.6


  " !$



Section 1.1

)

7

Fundamentals of Graph Theory

%            #     

Figure 1.1.7

 
 % !$

  " 

Vertex-Coloring

=  ./        "         
    $

          ;
DEFINITIONS



) %             ./
       $





  



) % ./         &        >


)
)

%           ./ 

 +,      
#      /#



 

 

"  + ,"   

REMARK

)
    
   " 
  "  
   
"  ¼ + ,"  #  #       0 1    /
  #

 

 0 1

EXAMPLE

)

 
 2   A      /    ./  
  
/#"    #   %"        #
/ /#    
/ /#B " ¼ + , * 

Figure 1.1.8


 %" +

, * ¼ + , * 



1.1.2 Degree and Distance
                   
.   
$ #     
     ÜC 

8

Chapter 1

INTRODUCTION TO GRAPHS

Degree
DEFINITIONS



) 
  +  ,   .     "  +,"  
#             #  / +2  
"  "        #   #,

  
 %               
           "  .      
   




) 
 
       9  #    .
   /    



)  
    .       #     B
 
   .    #      - /  
                 

)

% 

      .    6

EXAMPLES

)

2   C         9

 6"

" " " @" @" 
.  3
 

Figure 1.1.9


)


  " 
 #!&

2    6             

.
  
  
Figure 1.1.10















 
  !"
  " '"&   


FACTS

2       "  5DCC" Ü  7   #  .

(

) +-,                  # 
 

()
()

'  "  #            #

% /     
 9

         

  

Section 1.1

9

Fundamentals of Graph Theory

(

) '  "                # 9
 #   

(

)    9       " /   9    
     

(

) "  /   "    9      
     9   " #        
Walks, Trails, and Paths
DEFINITIONS

)

%    

    9      "

 * ¼ ½  ½ ½      
    *     "     ½              '"
"         ½   "    
 

¯ '    "  !  #    #     9 
 )  * ¼  ½         *     "     ½   
 &
¯      ¼ 
¯     +   ,   
¯ %       .         

)
)

 

)
)

        #    +    ,
%

!  

     .     .B  "  

%      

!         

%     
 + 
,

)
 )

)

 

!       

($


%             .   
%            
%  !"  "          .    

EXAMPLE

)

'      2    "  . 9       
 !      "   . 9            
   

Figure 1.1.11

10

Chapter 1

INTRODUCTION TO GRAPHS

Distance and Connectivity



) 

     
         
! #  



) 
 


     .    .
    !    

)
)
)

       

%    
 #           !
%     
        

%  
   
   .    . 
   !



)        .
  

)
)

          .

 

           

       .    

EXAMPLE

)

        2   
    B  
      #     

Figure 1.1.12

"
$ &&" 
   %)$ &&" 

1.1.3 Basic Structural Concepts
=      # 9     "    
     "     #       

Isomorphism

' "     
 "$      "   
        %  .//.   //
       
 "

DEFINITIONS



) % 
 
  
 
    . # & 
)
       "  .    &   .   
    +,   &  +,     '  "      
# &  
    +, +,









Section 1.1

11

Fundamentals of Graph Theory




) % 
 
     
       # &/
   ) 
   )               "
       &        # &         
&      +,  +,



)



=  
  

)








  
 


   

% 
        
 

         
  . !
! +  , *

*    

    .    

      &
6  



) %         
 
      
   +   ,       
EXAMPLES

)

     2            

           

% &


Figure 1.1.13
'  $  .   
     #      
9/ ! "        
0! &  !1
      &   )

!

*

     
 6
6

 
6





6



6

6

! *

     
 6
6

 
6




 6
6



6

= #                
   . !   . ! 

12

Chapter 1

INTRODUCTION TO GRAPHS

)

     2       "       ! 9 
>  ./#     

Figure 1.1.14

% &
 "" ) #!" *"



) 2    ;     /         9
+'             ,

Figure 1.1.15

%
 % 
 #!&  " 


    

FACTS

(

)      # # &    ./   
9  "+E, 

/.  /

()

%        !  + 

,"    !   /
          

( )

 #   "  #   "     9  
    (    "          
           "   # -. ;

(
)

-   +  ,    &  . 9    
   .
Automorphisms

          3        
 
DEFINITIONS

)
)

%    
         

               
       +, * 



)     
          
       + , * 

)
)



 



 

%   
          # 
%   

          # 

Section 1.1

13

Fundamentals of Graph Theory

FACTS

()
()

 . #      ./   
   #       /   

EXAMPLE

)

2       2    @"  . #     
 "     #             
    ./     /  

Figure 1.1.16

Subgraphs
DEFINITIONS



) % #    
           
+F"       #       #  #   ,




) '   "  
 

         *      "
 + ,"     ./"        
   
     "

 + + ,, * 



 + + ,, *    +

,       

    

)

% #       
 
    + , *  + , +%"
        #   "       ,
) %          #      # 
         '   "     (



 # 

EXAMPLE

)

2    2    :"      #  #    
# "       #  #     # 

Figure 1.1.17


!
    !& !
 

14

Chapter 1

INTRODUCTION TO GRAPHS

FACTS

(

) ? )   #    "   # #  #    
       #  #         +# ,

()

'   #   #    

        2 
*

#   #    

 " 

Graph Operations

                     
"   " 
" #          "   
# 
$  " 

DEFINITIONS



)     

       * +  ,"   
  "
       .          "      
+  .     #,



)    
      
* +  ,  
  .  #               
      



)     

  
 + ,    * +  , &   
            .          +  ,"
       +  ,



* +  , 



    # 

)    
  
 + ,   
           +  ,
) %  +   ,   .


)
 )

% 
    

    #  

 
           +  /
  ,     .   "           
        


)

   +


 ,    

+
+

)






   #

G  , *  + ,   + ,
G  , *  + ,   + ,      + , 

   + ,

 
 
  + 
 ,    

+
+

G 


 , *  + ,
 + ,

 , *  + ,
 + ,   + ,
 + ,



   #

        +  ,   + ,
 + ,     + ,  + ,"
              + ,         + , 
 + ,
 + ,     + $,  + ,"  $          
   + ,

Section 1.1

15

Fundamentals of Graph Theory



)                 ./
  /   &   "  "   ./    / 
  





)   

  %
   

   &   

   

 %  

EXAMPLES

)

2    A       / 

Figure 1.1.18

 )


+&""

2    C   &   

Figure 1.1.19


)

, "

2   
6      

Figure 1.1.20

-" !&"



) ' 2    C #"  .           
   /."       .    .   / 

1.1.4 Trees
                /
       "             
   ! %    3        


16

Chapter 1

INTRODUCTION TO GRAPHS

Acyclic Graphs
DEFINITIONS

)
)
)

%  
   +    , 

)

        #          

%     

   + " $$
$,

%         .
 

  

    9    

  
   + Ü ,"    $  $  
          "      .   
 "   + 2 ; # ,
EXAMPLE

)

      2   
  B       

Figure 1.1.21

"  "% +"

FACT

(

)         
#     .    ,

&½   &¾ 

+ Ü    

Trees as Subgraphs

 > #/             " 
    .     %   9    .  
  #  /         
DEFINITIONS

  
 2    '    "        '  
 

   
"             ' 
  

   




) %   
    
    '        ' 

 )

%
     

'

     /  

    #  

 

   

EXAMPLE



) 2    2   

"        '     #  
    #!"   /            ' "

Section 1.1

17

Fundamentals of Graph Theory

     "    "
" "          /
          ' 

Figure 1.1.22 " %" " 
 .
. .  

(#            2   

      ' "
   #             "    
       /        
FACT

() ? ' #      "    #       '    # 
  #        '    +2"        
       .    ' " "  /   ,
Basic Tree-Growing Algorithm

 #  /     . #  ! !     
       

 

  
 %




    /.   
//       6
Algorithm 1.1.1: /&

     

      

+%
%" 0"1 2

)")       .    
!")     '   +,     ./#  
'   3  '  . 
=  # 6  . 
'   3 #   )*
=   '       +,
         ' 
?  #              
%     .    ' 
=  #   . 
 )*  G
  '  ./#    +,

 +,

'

REMARK

) 53#!  " 4!"!"   +%
7

=            +      #!  ,"  
  %       #  9 +    "     
       
 ,   9      
 &
 "   #          +  ,   

18

Chapter 1

INTRODUCTION TO GRAPHS

              
         #!       

 

FACTS

   
"   #       #           
    +,
() '  .    #  /        .

() %         #  /      #  
 
Prioritizing the Edge Selection

  /   3 /      #     #  /   
Algorithm 1.1.2: 
+5" +%

)")    "    .    "
       3     
!")     '     ./#  
'   3  '  . 
'   3          '  
=  # 6  . 
'   3 #   )*
=   '    
F          ' 
?  #       '      
?  #  #       
%    + . ,   ' 
=  #   . 
 )*  G
  '   ./#  





FACT

( )
>      3           >  

)  "*+

 $  +

**+
*   ,"   *+

 $ 
++
**+
*   ,"      +

*$
   ,"   ,-
   
+$


**    , +  6 ,

References
5A;7   " 
" H/4 " CA;
5CA7  #I
" 
 

"   " CCA
5?C@7    ? ? !" 
 
"   -  " 
 4" CC@ +2  -  " J= J# " C:C,

Section 1.1

Fundamentals of Graph Theory

19

5DCC7 K ?   K D" 

     "  J"
CCC
54C7 2 4" 

" J  " CC +2  -  " % /="
C@C,
5 C
7 L    8 H   " 
 
  
 " K
=  M " CC

5667 =  " 

" # "
666
5=6 7
  ="   

"  -  " J /4"
66 
+2  -  " CC@,

20

1.2

Chapter 1

INTRODUCTION TO GRAPHS

FAMILIES OF GRAPHS AND DIGRAPHS


    
    

    !




 ' /< ' 

    N  


Introduction
=       "      O   
 O  9"    #     %     
   "     3     5= CA7 >   
   #         

1.2.1 Building Blocks
              #    #
                #  
 #   
DEFINITIONS



) %                &  # 
           

)
)
)
)
)
)



  

&

&    #  



      

&¼          
   &½      .    
          .   /

               

  

%      
   #        
                



(




  

&

)       /. 
 "       
 +N    "  #         #
  ,

)

   

   /. 



(

  "       

REMARKS



) %          0 1 "   
          /  #

Section 1.2

21

Families of Graphs and Digraphs

)

=  01   01    9      "
 0 1   0 1  !    

EXAMPLES

)

Figure 1.2.1

&"
   &" 


)
Figure 1.2.2

"
   &$&


1.2.2 Symmetry


   !         

Local Symmetry: Regularity


           
DEFINITIONS


)

%       .      

)
) % )     

¯ '      .     )

  )/    # 

FACT

()

%  (* 
  + Ü  ,   

EXAMPLES

)
)

2

) * 6 
 "   .  )/    

   

        ;       & +   6,"
    +  
,"     & +   ,

)

5= CA7   .   /    

Figure 1.2.3

 @  

 "% +
! 
 %" @ '"&

22

Chapter 1

INTRODUCTION TO GRAPHS

)

 &        &  &  
/    :/
.     ./   '  /   /  
  :/.     ./  



) (   /     
  

Figure 1.2.4

 A  "    ./

 6' +
! &&" 
 %" A '"&

Global Symmetry: Vertex-Transitivity

( ./        #    Ü@
   
      
DEFINITIONS



)     + * ,         *  
                   
  *  =      # #         #
#   * 
=                    # 
 "    B  
          
         # 

)
*

%      +B * ,         
    
       "    




  #

 .    "      
     &         )      >   ) 
   *  %            B   
              



)  
 +   " 
 ,   )/.
             & 







&

 

)     + +   ,   /!   /
  # +      , 6        
   
     

)



  
      )

 *



&

  G &







*


Section 1.2

)

23

Families of Graphs and Digraphs




    6/. /      2  
;

Figure 1.2.5

 5"


EXAMPLES



)  
   .    G Æ       /
   ' /!        & 

)

%      /!      J   )  /
 "  #"   "    "    




)  J   ./  "        
     +   ,     '      
          6" "            
 "  "    



)

         & 

FACTS

()

54#  3  7        # 
9             &    9 > 
.       + 

()

=    /   # +  "     
   )

+ *



&
+


&

 * 6
 

1.2.3 Integer-Valued Invariants
           #  /  /
                   
# (  #          
Cycle Rank

    $$
  6      "     
 
+   ,

24

Chapter 1

INTRODUCTION TO GRAPHS

DEFINITION



)          * +  ,   #    G  +
@       !   !    , 8 " 
 
 + , "   !   #  + ,   + , G + ,
EXAMPLE

)

     ! 6   

Figure 1.2.6

 " %" ! " 6' '"&

FACTS

()



5  3  7      9   
'     + "  5DCC"    7,)

'
'
'

(

   + "

     ,

   
        
%      '   # .  
   

) 5'   + ,
    7 ?  #      
  )
+ ,
+ ,

&½   
' '    '  # #  #    
 ' "  '   
¼

.  &      .

¼

        
+           
,

()
()

  !          !   
%           

Chromatic Number and k-Partite Graphs

'       "            &" 
"    &     >       
/            .   ;  ;

DEFINITIONS



) %              #    
   + 

,    "     &        
 +2   "       &½      , ' ,  $ 
       "       #  ,$  

Section 1.2

Families of Graphs and Digraphs

25




) %        #        .
       &            '     
      ,  $"       &


) %   )      #     )  + 


,         &         


) %  )      )/         
  &        >    %    
  
 '  )       ½      " 
    &½ ¾


 "          ,"  &
 
EXAMPLES

)
)

-   #  

-     #     #  "   
 #  #  

)

 

  /   &       /    

Figure 1.2.7

 &" +""
 &  .  *     

FACTS

(

) 5     3  7 %   #      
          + "  5DCC"  ;7,

( ) %   )/#       )/ 
(
) 2 )  "  #        )/   HJ/
k-Connectivity and k-Edge-Connectivity

  #   3      $$    *$$
             :  . 
     
DEFINITIONS



)        "  - + ,"     
#         /     



)  
         "  - + ,     
#          / 

26

Chapter 1

INTRODUCTION TO GRAPHS

   #  0 1          
- "    -  .     -  - "  

)

% 
    -  )    )  
 -9 "
 )/     )           /
      



) % 
  /   -  )    
  

 "    )         )/ /    
 



) %  
  )  
+ )
 ,       
     )          



) %  
  )   
+ ) 
 ,    
        )         
Minimum Genus

  #   3         
DEFINITIONS



)     
+     
,    
 
 #   
 #      #  / + : ,
   /  


)

%    6   

1.2.4 Criterion Qualification
%                  
   "  " #             .  
DEFINITIONS



) %         !       .
 +            
  .   ,



) %            +       
     ;   .   ,



) % )/      )       # 
        + ; ,



) % )/     )  
    
   .    +  ,

 /



) % )/ /     )
  
   / /
          +  ,



)     0+ ,            B    
 0+ ,   &              .
%"       #       .   
   
   0+ ,

Section 1.2

27

Families of Graphs and Digraphs



) %             .   #    
   + Ü,
EXAMPLE

)
Figure 1.2.8


  " 


FACTS

()
¯

5?    3 7      9 )
    

¯ 5L7      #      #    
  .      
¯ 5:67 H       2  
C     #  

Figure 1.2.9

()

   !& !


%           

EXAMPLE

)

Figure 1.2.10

 "!" %"  " ! '"&





28

Chapter 1

INTRODUCTION TO GRAPHS

References
5:67 ? =  !"  3     "   
 C + C:6,"
CP ;
5DCC7 K ?   K D" 

     "  J"
CCC
5L7 K L3"
Q   R IQ  =    I."
    ;6 + C," :;PAC
5= CA7      K = "   ! 
" (. F   J"
CCA

Section 1.3

1.3

29

History of Graph Theory

HISTORY OF GRAPH THEORY
  


 !" 
 #

 #  


   
    
;  %  


Introduction
%             A:A" /  
 #  #!  :;  ? - + :6:/A,      
 LS
 # #  #     3     
          2    #  
5 ?= CA7  5= CC7

1.3.1 Traversability
       #  #!  -T !   LS
 #
#  # + :;,"   #9       
   "
        #     J  L ! + A6@/C;,
   =     4  + A6;/@;,       
  "
The Königsberg Bridges Problem

 #.

   
" 
"    2   " !     /
  !       #   LS # .  O  
"    !  #    Ü
  .     
  
   "

C

c

d
g

A

c

D

a
B

Figure 1.3.1

b

 ' 
  78




30

Chapter 1

INTRODUCTION TO GRAPHS

FACTS 5 ?= CA"  7

() (
@ %  :; ? -     0     #/
         1   %       J# "
 "          !    # 
() ' :@" -            " /
             #          
  
() - 5-) :@7        #   
 

"
 
 #
 
     0  #   /
         1 %   :@"     
: "     #          
  :;

() -T     
 " 
#  #"       
  

       LS #
     # 

() -           #" #  
 #         9     %" " " 
+ 
,      %  %   &   +     
#  #  %    #  %  ,"     %
" 
"  

 & &  4   #         9 . "
#     LS # #  #    
() '     #" - #   #  # 

  .    %" " "          #  # 
         /     0 ! 
1"      ./       9     # 
 

() -T              )
¯ '        
 &   #

    #  #   "  

¯ '  #  #     .   "   &   #
          
¯ '" "          #  #   "  
9  &  #       
         
 / "  

       " 

() -    "    #       "  

   #"            # " #   
%             #  4 3
54 ) A:7  A:
Diagram-Tracing Puzzles

%          * $ "%%

"    9  
          # #   ! 
33  #  #!       P  ."    
%  .

Section 1.3

31

History of Graph Theory

FACTS 5 ?= CA"  7

( ) ' A6C ? J  5J) A6C7           
     #)

 

           "   9     
   $. #        #      # "   
       &         9     
  
J         #    #      " 
             #  '  
  "  9             
 

(
) (  /  33     #   5) A7
 K  ?   5? ) A:7      #! $ 
   
"
      0 1    
() ' AC" ( 9 !   #        
           #      #   
     "    # 8   5) A: /7   #
 

&

()    #   LS # #  #   / 

33     3       C  '     #
= =   5) AC
7  
 %

  #
   
   #        2  
   #
C
g

c
d
e

A
b

D
f

a
B

Figure 1.3.2


  " 78


 

Hamiltonian Graphs

%    #    #    #   
      &     .      
4 T $"      
  "
+ Ü;," 
  & #    L !"       #
   4 T        "   # 
FACTS 5 ?= CA" 
7

() %  .    #   /
 " 
"    
   ! T    # "        @ 9 & 
           #  #  #!    

32

Chapter 1

 "     
5<) :: 7  

INTRODUCTION TO GRAPHS

  # - 5-) :;C7" %/ <  

() ' A;; L ! 5L ) A;;7            

             &  4   
   /      #       " 
   .    #  # 0         #1 +
2  ,

Figure 1.3.3 7)9 :&   ; 1

() %      !  /   #" 4    

             4 #9   "
     + 2  ,"          
       "      .   
R

Q
P

Z

W

B

X
H
J
V

S

C

G

D

N

F

M

K

L
T

Figure 1.3.4 <"9 &


() ' AA" J       /       

    
2  ;,

Figure 1.3.5

 #9   # =   5@7  C@ +

!""9 +'" +" $

() Æ         #      #  #  %

 5
;
7" ( ( 5(@67" K %    < I
  C:@"  

Section 1.3

33

History of Graph Theory

() 4      #   " # %  /4 + C@6,"
4 8  + C:,"  

1.3.2 Trees
    "     "     
 !   L > + A
/A:,"   /    
        ! ?"    # % 
+ A
/C;," K K  + A6@/C:,"  JI
 + AA:/ CA;,"  " 
    #           
Counting Trees

-   9           #  
>  " #     #        
  "                  
  -     !        Ü@
FACTS 5 ?= CA"  7 5JA:7

( ) =  !    #   #  !    0 > 

        1"  5) A;:7     /
    

(
) T    !       " # #  
 #     + 2  @,
root

root

Figure 1.3.6 =""
 " "

?  ! #  #      #"    
   
G !½  G !¾ ¾ G !¿ ¿ G   
 9    
F    9 " 

+  , ½  +  ¾, ½  +  ¿, ¾    
 #    # !     

() % A:6"    K       $ 0$
  $ 0$    
() ' A:"  5) A:7         Æ #
        "         "
                 
!    

34

Chapter 1

() ' AAC"  5) AAC7    

INTRODUCTION TO GRAPHS



   #  #
     4 .         * @" # 
            # 4 JS
 5J A7)  
   #   //   #   #  
9     
   # 
     

() '      C:" JI 5JI:7 #      
            #      
#                   
     JIT !     # K 4   5
:7" #
  T   #    $      #&
() ?           #  ( 5(A7 
        + 54J:7,  #9 
 # 2 4 54;;7"    5@7"  
Chemical Trees

 A;6     !      #   .  
       +,  " +,  ! " # 
       #     # %  
 "      #   # #  "    %. 
             2  :
      "     "     


H

H

H

C

C

H

H

H

O

H

H

H

C

C

H

H

O

H

Figure 1.3.7 ""  "
FACTS 5 ?= CA"  7

()   T     .        

  " #  .      + ,     
 # >     2  A      
  
H

H

H

H

H

H

C

C

C

C

H

H

H

H

H
H

C

H

C
H

H

H

C
H

Figure 1.3.8

C
H

H

H

% > !"  !"

Section 1.3

35

History of Graph Theory

()  5) A:7  /      Æ +!,

  
# "           B   
#   #    !   *      A
2 
H#




      




;

C




A

() = L  >   #       #   # 

    #  9     "        
        #  ' A:A"  5) A::/A7  
   &
#     " !  )
-       # . # #   "  
    L!I      
 

     

  "   /  

( ) ' A:A"  5) A:A7            
             
  !

 "        
(
) ?   

            C
6 
C6 %  ?  K L   5?
C7   3     
    "  JI
T #/       #
      

1.3.3 Topological Graphs
-T    5-) :;67         
"          '   .    
   ' C6"     3    # #  
   # L3  3 L ! + AC@/ CA6,"   ! P # #
H  #" J "   P  .       
  
Euler’s Polyhedron Formula

 !           " #       
!      #   #    "    "  1 
    )



 G1 *

'  : " I
    "   #   
  -T    #    4 "  
   
   "   #  !     

36

Chapter 1

INTRODUCTION TO GRAPHS

FACTS 5 ?= CA"  ;7 5CC7

()            "  
H# :;6"  -    #
   #  "   
+ ,  &  + , # " "  "
'    #         #    
#     .  #    #   "   G  *  G


() -  #      ' :;
    # /
 " #           # %/8 ?   5?) :C7
 :C"           
() ' A " %/?  5) A 7 #     -T  # /

  &                 
    

() %    " /%/K ?   5?) A 7      

      .  "          
#             4     
     -T    P         " 
     "    /        + "  
  RT   , 2    /  " ?     


  G1 * 6
 .        "     #      
        " 

   G 1 *


 #      
& 
 &$"   9 

  
 1
 $ $ 
$ + : ,

() ' A@ /
" ?   5? ) A@ /
7  
 
  '
 
(
"  ./

      ."           >
  3  #  -T    !   # $   
#9     '  " 4 J I !  ?   T 
     AC;/ C6         #   

()  !  J IT !   "       /

 # 8
  J 4  5
46:7      + ,   /
) '

 
 
 *
 
!
 4      #
( <# 5<

7      9          % 
8     C @

Planar Graphs

             #    /
  &    #    &    +   2 
C,   #    #   "   #9 
# L ! 

Section 1.3

37

History of Graph Theory

K3.3

K5
Figure 1.3.9

 7!"%)
 & 

& 


FACTS 5 ?= CA"  A7

() %   A6" % 2 8S#      33    /
)
    !     '          
      !          #  
                 
        #   U
  9  !        #      
       #      "  
# 4  3)
 !        &           
!   #              # U
'     "  #        

&  

() %  #"     #"   

" 
"  
* *

$ $ " 
"   # 4
  5
 7   " 
 C )
 33     " "    "  =" "  -"   
  " %" "  "        + 2 
 6,
  #     



&   


W

G

E

A

B

C

Figure 1.3.10


+%"+&"&"$ 

( ) ' C6 L ! 5L67 #         
  #        B     #    
# ( 2 !  J %  

&

&

(
) ' C 4 =  5= 7   #        
 #     

           

38

Chapter 1

INTRODUCTION TO GRAPHS

 4  "          "      
     #     #  #  8?  

() ' C; =  5=;7  3         

              .    
     "   5;C7      / C;6 
#   L ! /             + Ü@@,
Graphs on Higher Surfaces

%             +#    & ,
                      
            !        " 
 .    " # J 4  + A@ / C;;,  ? 4> + A@
/
C@
,   # "  # 4    3 + AA6/ C@,  / #
" #  #  #             
 D    4  &   C@6  H  #  J
  3 L ! T       CA6
FACTS 5 ?= CA"  :B  :7

() ' AC6" 4  54) AC67   #     

&

   4                
   # #  " #           

() ' AC " ? 4> 54) AC 7    #     

 #        "     4  T
     #         

() ' C 6" 4  3 5 67 .  4>T       / /
# "    8S
#  #   &  "    /
  4   4  #      L  #" #  
   # J 2!  527  C"       .  
  ' C;" ' L  5L;7       / #
 " "  @
()  4     / #     C;
#

       #    #   Æ"  
66       
  8        /
C@6"       C@A #     D  5 D@A7"  
=  T 5@7 #       C@   $   "   "
   # K ?  5:7       3 #  
        #&"    3    
  "
+ Ü:,"        +    = ! 5:7,

() '  9     CA6     " #

  5A;7  "    #  "    0# 
# 1    + Ü::, 4 "    "  #  # 
#      "     2 / # "  
    "   C:C 4 4 " K J 4!"    = 54=:C7
#     6 #  #    &  

Section 1.3

History of Graph Theory

39

1.3.4 Graph Colorings
- !                 " 
 "  # / #      # 2   
 A;
"   # + , 01 # %  L   A:C
'    # L %  =  4!  C:@" #   
   !  L"    !>" 4   4"  "    
  #9   # H  #"
   " J "
 #   5 CC7 8  "        # 
                 
            "     
   5 AA67"          <  < 3   C@ %
   "    #        
 #     D   C@A
The Four-Color Problem

8      #  #!      /
# / #      
FACTS 5 ?= CA"  @7 5= 6
7

()    !      / #     
%
 8   4 " 
 (# A;

 8   #    
 !       #   &       
 #      >         
2  !  "       #    # 2 "  
            -   4      
#
()
 8        "     #      #

  Æ   ( 6 %  A@6"  #     "  
  #!     

"   #
 8     
 
  F #   J "     /    #

( ) (  K A:A"       ?  8   " 
!   #  #    "  #    
 #    Æ    "        Æ    
T      

(
) ' A:C" L 5L) A:C7"   #     " #  
    /    
  ! 
 "
    #   #  L      
         #"          
     # .        4       
 9"  !    #"*$  "         /
         LT         "
  "  9    " #         +   
    /  ,   
() ' AA6"   5) A:A/A67  0  1   / "
     (     / #     

40

Chapter 1

INTRODUCTION TO GRAPHS

  ?
  +?  ," 2  +   ? ,"   <  
  K 8 = 

() ' AC6" 4  54) AC67 #      +
  ! #

  
 
 "        LT "    
    / "   3   #    "  
   + Ü  , 4  #9 #    .  
 #"          C6  L      " # 
 #     

()
      
6       "       

   LT        

 O      "
           F #     #
J = ! 5=) C67 + 2   ," # J 2! "  # 4 ?# 

digon
triangle

quadrilateral

two pentagons

pentagon and hexagon

Figure 1.3.11 ?&)9 !' "

      $
 $+  O        
             # .     /
 )          /.   /
  !> 5 ) C 7            2 

+!     !>  ,     #   

Figure 1.3.12

 /)* 

() ' C
"  !> 5
7    #        
 ) "               )" 
$ $ "

  

 

() ' C

" 2!  52

7    #     #  /
 "       /      
    #      

  
;

() % C;6 4       #     # /
   (  .  " 4 54@C7        #
  

Section 1.3

41

History of Graph Theory

() ' C:@" %  4! 5%4::" %4L::7"      K L"

#    #   A
  #   " #    /
      9  #          /
     #  

() % CC" #"  " "   5C:7 /

      F        #   #
     #        "   3  %/
4! "   #    #   @   #   

Other Graph Coloring Problems

%    !   / #"  
#           

 #      

FACTS 5 ?= CA"  @7 52 = ::7 5KC;7

( ) '   A:C       " L 5L) A:C7    
#                 &  
  >      /    !  #
4 =        C
 #  #9 !  
/ #

(
) ' AA6"   5) A:A/A67    /   9  
            # 
         .

   

() ' C @"
 LS 5LS @7        #   
.     # 

  + Ü

,



()               &    
>           C6"      !  = "
    J
        
() ' C " ? ! 5 7      #     

 .       G "
  + Ü; ,

 9       

() '  C;6" #     ./      #  %
"
       $ $
 "

() ' C@" <  < 3  5< @7           

.       #   G  '    "
< 3         /  

()       #   /  #   
 #  3 #  #    O  ." 8 3  
        C@6"  J - V
         

42

Chapter 1

INTRODUCTION TO GRAPHS

Factorization

%   )* 
          )     
#      # "     ./      %
)*&$      )/  #            
 2  !         # K  J
5 AC/ C 67  =   5 C /
66
7 + Ü;,
FACTS 5 ?= CA"  67

() ' AC " J 5J) AC 7         3  
  "      #       '    
   )  "   )/    #   
/ 4 
   /     /"        
  01B     #  &        #      
() ' ACA" J 5J) ACA7         "  

 
  " + 2   ,"    #     /B 
"  " #     / + !,  
/ +   
 ,

Figure 1.3.13

 5"


( ) ' C:"  5:7     3       

/ 2     .        3    
   )/"   )

1.3.5 Graph Algorithms
      #  #!   C "  2 
                  
  3 + Ü
, 
6           #
      #"       #"  

 
  
 + Ü,"      #  #    
   '    #      !"   
"     + W .,     #    #"   
      !   
FACTS 5
A
7 5??A;7 5?JA@7

(
)  
 2

  
"        !   

   #            "     

Section 1.3

History of Graph Theory

43

 A  '          C6"  J "  
  3   %H
            

3 "
  2!"   8 K 5
2K;7       
    #  C    '  CA6  # 
C
  
  # J #    5J A:7 + Ü@,

()          $$ " 
"     ! 

  /         "  #  #!  ( !
5
@7      # K  L! 5L;@7 %    "  
< KI ! + C ,"    #   J  + C;:, + Ü 6 ,

()       #  
3 
  2! 5
2;7
    .  $     #         
 !"  #  -     4 54@ 7     ( 3

  /   !
() 2      "     "       !   

C6  C;6"  1 +J  -      9,  # 
F H  #    #    #   J8 + $

4,  # 
 J  H      3    
 & + Ü
,

()    Æ             
 !" 

   # !     - =
&! 5
;C7 + Ü 6 ,

()     #"         

       "    # 8   +8 /L L , 5@67
 C@6 + Ü,

() '      #         /

   #  &#      9     !   !
 LS
    #       J   4 54;7" 
!    0   1 54<;67        #& 
  #     #            
    + Ü ,

()    C@6  #    #   #  Æ/
  "  -   5- @;7  #      / 
   .  ! 5: 7" L 5L:
7"       
HJ/   "  "       / #/
     "

* $


J"        4  
 #  HJ/  '   !   J * HJ 2   
#   5K:C7

References
5%4::7 L %  = 4!" -    /#) J "
  "
  

+ C::," 
CPC6
5%4L::7" L %" = 4!"  K L" -    /#) J
"
  #  "   

+ C::," 
CPC6

44

Chapter 1

INTRODUCTION TO GRAPHS

5 ?= CA7 H ?  " - L ? "   K = " 

 ,-./0,1./" (.
F   J" CCA
5
7 
  !>" %      # 
"  ! 
  + C
," 
P@

    

5 7 
  !>"    #    " 
  
 ; + C ,"

;P
A

5 ?@7 
  !> 
  ? "    "   
 

" @6 + C@," ;;P; 
5
@7 ( !" ( & I #I   I
I "  " " & 2

+ C
@," :P;A
5 7  ? !" (         !" # 
#
  "
: + C ," CP C:
5) A 7 %/? "    Q /  I "  )
#0


 C + @, + A ," @APA@
5) A;:7 % " (         " #
  +,
 + A;:," :
P :@
5) A:7 % " (      " #
  +, : + A:,"
P@
5) A:C7 % " (     " # % 
 " +
+ A:C,"
;CP
@ 

,

5) AAC7 % " %   " +  #
  

 + AAC," :@P
:A
5: 7 % !"  .   /   " ; P ;A  # .
  "  
 !  " %8" H D!" C: 
5) A7  " 5   7
         #"
  &

+ A," 
6CP
@
5CC7 J   " #
" #  F   J" CCC
5
A
7  
3 "    #          " 3

%

 + CA
," PA
5
2K;7  
3 "
  2!   8 K"      /
  / #" 3
 %

+ C;," CP 6
5
46:7 8
  J 4  " %   " ) '

 
 

*
 
!
 + C6:," ;P
6
5
8) A@67 %
 8 " %        "    
   " # = = "

" 

 H @C + A@6," ;6 P;6

Section 1.3

45

History of Graph Theory

5
;C7 - =
&!" %     #  .   " &
 

+ C;C,"
@CP
: 
5
;
7  %
"    # " #  
 " +,
+ C;
," @CPA 
5
 7 4 -
 " J.  " "  @ +K C ,"
C ,"



6  +% 

5- @;7 K  -  " J"   $ "   
 : + C@;," CP@:
5-) :@7 ? -" + :@,   #         " 0

 

"
 
 #
 
A + :;
,"
AP 6
5-) :;C7 ? -"   T 9    9      Q 
"
  " 4
 ; + :;C,"  6P:
52 = ::7  2     K = " )
0 ! 
" J " C::
522;@7 ?  2 
  2!" 8.  $
 
 A + C;@," CCP6

    !" 

52

7 J 2! "    #" 
  
  + C

,"

;P
@
527 J 2! " %  .  #"  
 #
   + C," @P@C
5K:C7 8   
  K"  
    



 ! &#0 


" = 4 2  " C:C
54F=:C7 4 4 " K J 4!    = " 6      #
  &  "   
 +,
: + C:C," 
P:6
54@ 7  -     4" 8 /   ! $ " "   

 C + C@ ," ;; P;;@
5:7" K ? " <  "  


 C + C:,"
CP
@
5:7 K ?    = !" N    )     
4  #" #5  
 ;; + C:," C P6

5@67  8 "            "  
 "
6 +
@/
@@ * C@
,B 



 + C@6,"
:P
::
5@7 =  " ( # #    " 4 
 
 " @C
+ C@,"
:
P
:;
54;7 J 4" (    #"   
 " 6 + C;,"
@P6
54<;67 J  4  4 - < "     #" 
  
 :
+ C;6,"
P
;
54) A;@7 =  4 " 8     
#
  +,
+ A;@," @

     "

46

Chapter 1

INTRODUCTION TO GRAPHS

54;;7 2 4"  #   "  "  "   "  

 
 " :A + C;;," ;P@
54J:7 2 4  - 8 J" 

54) AC67 J K 4

PA

)
" %   J"
 " 8/ " +  #
  

 +

S
54) AC 7 ? 4>" F#
 J#  H# # "
::P;A6


 

C:
AC6,"

A + AC ,"

54@C7 4 4" F  3 < ##" 4  

  
"
A 6WA 6WA 6#"  #    ' " 8 /< /XS
 " C@C
S
54 ) A:7  4 3" F#
 8S
  ! "   ? 3  =   
 F# 3 " 
  @ + A:," 6P

5KC;7   K   " 
 #
 " = /' " CC;
5L;7 ' H L " %    4   "

AP
 

 
 #
 

 + C;,"

5L:
7  8 L"   #    #   #" A;/ 6   
(
!  
   +   - 8   K = ," J J"
C:

5L) A:C7 %  L" (     #   "
+ A:C," CP
66
5L ) A;@7  J L !" (      "
 @ + A;@,"  P A


  


#
   % "

S
5LS
 @7
 LS " F#
   %   
   
8 " 
  :: + C @," ;P@;
5L;@7 K  L!" (     #       
 #" # 
 
 " : + C;@," AP;6
5L67 L L ! "   #Q  #    " 
; + C6,"
: P
A




5??A;7 - ? ? " K L ?" % 4    L" 
  
+ ," 
2
 "
 #
  
  


3 " = " CA;
5?) :C7 % 8 ?  "

)6

 
6
6

+

  ," 2  
" J " :C

5?) A 7  ?  "
I  I  T IQ   T- 
 Q "  .    IQ   #" 6
    "
" #6

   + A ,"
: P6 
5? ) A:7 K  ?   " <  3  "
H  %# + A:," A PA:;

'
 "
 +%#

, 8

Section 1.3

47

History of Graph Theory

5? ) A@ /
7 K  ?   "
  S  ." 
 7 
 *  '


  6 + A@ /
," C:P A

5?JA@7 ? ?I3  8
 J" 8  "  !  


0

C" H/4 " CA@
5?) AA
7 - ?" %6
6
 
6
8
" < "  /< " J  + AA
,
5?
C7 %  ?  K L  " '     "  #
  
 
+ C
C," 6
:P 6:C
58@C7 K 8" ? #Q  I          #"
  
 @ + C@C," ::P C;
5(@67 ( (" H  4     " 
 
 
 @: + C@6," ;;
5(A7  ("  #  "  ! 
 C + CA," ;AP;CC
5J A:7 8 = J #     " (  3    ;
/     
 # # #  " 3
 %

 @ + CA:," P:
5J) AC 7 K J"
     S
 "  


6

; + AC ," CP

5J) ACA7 K J"   IQ   " 
6
 
 ; + ACA,"

;P

:
5J) A6C/ 67 ? J "       Q "  )
#

  + A6C/
6, + 6," @PA
5JI:7  JI" L#   %3#   S
 "  
  <#   "  
 @A + C:," ;P
;
5JIA:7  JI     "  )
 !  9 


   "   " CA:
5J A7 4 JS
" H      3 
S# J " 
 
 #
 
+,
: + C A," 
P 
5@7    " (  #  /   "  0
 
 " A + C@," CCP 6
5
:7 K 4   "    /   # " 
  
 C
+ C
:," P;;
5) A: /7 8  " -   #  #   9 
A I T
&      # TQ  Q    &"   #  
+
, ; + A: /," @P
6
5 :7    "   

"   " C:
5 D@A7      K =  D "     4  /  #/
" # &  " :" @6 + C@A," AP;

48

Chapter 1

INTRODUCTION TO GRAPHS

5A;7 H #  J
 "    O  "  "2
 
 ,1;< +  ' % ," ?  8  ? H   6
+ CA;," #  F   J" ;P : 
5C:7 H #"
  " J "   "  /
"   
9 "
 4 :6 + CC:,"
P
5) AC
7 = =  " 
 %

  #
 ! #  
#

 
+   
 %

  )  ," 8 "
? " AC

5= AA7 4 " 8  # 3   K = " %    ) -T LS
 /
# "  


+ CAA," P C
5C7  - " %           !"  
 #
 
A
+ CC," AP ; 
5) A::/A7 K K "     #" &
: + A::/A,"
A
5) A:A7 K K " (              
          #  9 " 
  

+ A:A," @P
;
5) A:A/A67 J   " !      " # % " )

6 + A:A/A6," :
C
5 67 4  3" -   !  
S#  J#  LS#    /
 2S" 


 
 
0$
 C + C 6," ;;P :C
5@7 =  " (      "   
 "
+ C@," CAP 6 
5:7 =  "   3     "   
 "

+ C:,"
6:P 
5;C7 =  " 8   "   
 
 " C6 + C;C," ;
:P;;

5:67 =  " (          "   

C + C:6,"
ACP
C@
5<) :: 7 %/ <  " 9   #Q    " 6
 
" =# > + :: ," ;;@P;:
5<

7 ( <#"   " " % 8  9 ? C @" H D!" C


5< @7 < < 3  " (           / "  
 
 + C@,"
;P6
5< @;7 < < 3  "        "  
  ; + C@;," CP :
S
5=) C67 J = !" F#
 !   < #3" 
  ;A
+ C6,"  P
@

Section 1.3

49

History of Graph Theory

5= 7 4 = " H/#   "
+ C ,"
;P
:

# &  " :"

5=;7 4 = " (  #       "
;: + C;," ;6CP;
5= CC7  K = "  "  : 
-   " CCC
5= 6
7  = " 

:


  


  !   +  ' 8 K,"

 "Æ
" J  F   J"
66


50

Chapter 1

INTRODUCTION TO GRAPHS

GLOSSARY FOR CHAPTER 1
&$&&
)     $$



 '"1     * +  ," 
  
 #,

 ./




  )        
    /   +  .   


 
 +
 ./



,    * +  ,)        
  /     +   ,

@&&$ "1 P     


  . !

 





   ./ ½  ¾    ) 

      &
6  
@&" 
)            

! +  , *

@&" '"&)               
&)   $ 
""!")       B   (      
!" P   )        
""
) 
/  B       #      
         &          +2  
"   &½      # #  ,

!#!"  )  + ,    .   /
&" !&" + " $, P       )  

+


 , *  + ,
 + , 

+


  

 , *  + ,
 + ,   + ,
 + ,

 + * , P         * )   
./  "              
  *                 
           

-$$
½

-$$
¾)               


 + * ,     #  0 1

&"  
)  $
 "        
&" '"1 P   )  .     9      
&"& ! P    )   #     

/#

&&!"
  +B * ,)          
&)   #        #B  $
 
+&
)       ./      
&
)        +      ,    
           01B   01
       #
" )          +  ,     
 &   +"  ,      

&")    *$"


51

Chapter 1 Glossary

&" ""
)    #       .  
     &   .       '      
     ,  $"       &


&" 
 & )           #    
  "      #  

&"
 & )                 
 &  #   

&" )+""
 &½ ¾


 )    )/        

  &        >    %    
$"
 
"   "


&" P   )  .   # 
&&"
)     #           !
)+&&"
)             )   
  "    "    #      

)    )/  ' 
 -+ ,  - + ,
&" P    ) +  ,   #      
&&"'"$  
 )    #

&"&$ )+&"&
)      # )    /
#

     

   + Ü; ,

&"&$ )+&&"
)      
   .

   + Ü ,

)

   

&"&$ )+
+&&"
)     /  
  

     

   + Ü ,

)



  /

&!
)  " $  "
&!)  " $
&!"+
)        #     
&!"")  $* (
&!"+'"1 + $",)  .     #  
  

&$&)         
&$&
 )  /.     "            


&$& ) P   





+ , G + ,

* +  ,

 + , )  #  + , 


 + 
$, P   .     "  +,)  #  

          #  / +2   " 
    9  #   #,


 #!& P   )  9  #    .  
 /    

"
 '"1    

* +  ,)        
. " #                  
  

52

Chapter 1

INTRODUCTION TO GRAPHS

* +  ,)       ./
 " #     /           

"
 
   

" P    )  .  $$ $" !    

 +  $  ",)           
  )             &   
&" "&   .  "  .  P   )     
 

  
   "  

! 

&" 
 +  $,)

         
"
            '      "  
      

&"
)   "
&" P    )        

"& P #     )       ! #  
&&"&"$   '"1 P    )     .  



)  #          " * +  ,

+&
P    )       / 
   $



  






 ""!")       /      #  # " 
  $  


+&" P      )  
   ./ 
   

"  

+    , 
       

     

)+
+&&"
)          
         #      


+&&"'"$  

  '  

-+
¼

, 

)    #
- + ,

)

  )  

 



)/ /


+""'
)              
"$
 & )          
")   "
!
)      !       . 

+               .    
  ,

! " P   )            
&" P   )    # 
)+&" P   )  )/    # 
")        
" 
 P        )            
         B     
    

     


)        #   /   /
    .   .  

53

Chapter 1 Glossary

!)   

 * +  ,)      $
     
"     
             "

     "


"
)         +
     Ü;   .   ,

Ü      

)   $ 
$&!
 + )  /!   /   #
$&!   )  /    +½     , 6   ,
&&)     #           

 P   . )  #          
!& !
 P        * ½        )  # 
 ./



      

     



'" P  
 "
  
" '"1 P   )  .    6
&
)    #         H/
 )


* 

 '" P  )          
     B  "        #

  )     # &   )    
 )                 "       
&        # &          &     

 


+,  +,

  )  # &  )      
"               

  

        
   +, +,   



@½ +

"
,)       
 

+
+

       

G  , *  + ,   + ,
G  , *  + ,   + ,      + , 

G

   + ,

@¾)   G     0& 1    
@¿)       0& 1    

"   %))  #   

½ P    )     0+ ,  ./     /
  "           &     
    .



  


¾) %      # 
  "   .   

 

     0+

,

)  

&*
"

)    

&*
"

!
! +
!,     )   #   
 #      # 
H ) 2 + ,  2 + ,

/ + :

,

   /  

54

Chapter 1

INTRODUCTION TO GRAPHS

1
)   " 

  $  "
!"+&)                  
!"+
)          "        
!"
)       /    .  " #     /



 P   .)   & .
" 
 P      '    )         ' 
!
)    ./   /  B   &¼ 
&"
  )   /   /  &¾ 
"   '"1  P    )             
  

 +, * 

P    )              
 + , * 
!"
 P   . )  #          

"   

 

"$ &"
)     #      
3+""
)    ./        3 # ½    
+  " 

,"   
#

     &        

"" ")  3*"  
")  
        .    
" &)  $

 
        .    
" )  " 
       .    
"
 ()  /.      "        
      +N    "  #     
  #   ,

(

5"
)  6/. /  "      ;/  
    "
  

  / &             


)        6" "     #    
  

      

"&
)  !         
"& )       /     O  " #"
 "  "  

!&")  $ 
 " $
 
)     &       
 
+&
)   /        &    
 

 '"1+&
)  ./        &   
  

! P    )     $$ $" !    

!
)       .       '  )* 
 
 .     )

55

Chapter 1 Glossary

+)     &          
+!. + P    )    &    % *

%     

   





 
)     /    /

)      /   / 
1)  .     /  Æ       -  
'   )*
"
( 

)" +
  

/ * ) G 
*

,   )/. & )           

&


!
 P    )  #      .  

" P    )  #     
" A'"1B 
P       )  #   #  # 
# 6     

 #  #

    

"
$ &&" 
)       .   
.    

!

     



!
  
 )      





 

"   

!)  -
")   $ 
")  !         
")      
" 
 P      '    )        ' 
"'
 &½ )     .    
"' %). ".  ")  !    3
!$

 P     .     #)    
        #  + "      /  ,

  )        ./
   /   &   "  "   ./   /
   

! " P    

'&)    
'"1 ""!")      ./      #  #


'"1+&
P   

)      ./

   $





  





'"1)  #          " * +  ,
'"1+""'
)              
%) P    )    9  * ¼  ½ ½ ½       "  


 *     "   

" &) 
" ) 

!

!

½



           

  .         .

  .  >      .

%)$ &&" 
)    
  "   
%
" P    ) +  $
,   #      

Chapter

2

GRAPH REPRESENTATION
2.1

COMPUTER REPRESENTATIONS OF GRAPHS

  

 
2.2

THE GRAPH ISOMORPHISM PROBLEM

  
    
2.3

2.4

THE RECONSTRUCTION PROBLEM

   
  


RECURSIVELY CONSTRUCTED GRAPHS

 ! ! 
  

  
     "  
  " 
     "  
GLOSSARY

Section 2.1

2.1

57

Computer Representations of Graphs

COMPUTER REPRESENTATIONS OF GRAPHS

  

 
  

  

 
 
  

 

 
 

  
   
 
 




Introduction

  

   

    
  


 

 
    


  
  

 
 

     !
  
    " 
 
     

 
    #  #
 
     
 
 
 

   
 


 

    
 

$  

    % 
 
   


 


     
  #    
    
 
     
    
     
  
       

2.1.1 The Basic Representations for Graphs
 #  


  
 
  
  &

 
'  
&

  
DEFINITIONS

(



 



(

  
  ) *  +
    ! ,    
   

   -
    

 
* +  





 


  ) *
+
    ! ,    
      -
     
  







 
( .  

 
   ) * +, 
'       
'   * +  
      
 

 &
   
 
   
( .  

 
   ) * +, 
'       
'     
       
 

 &
   
    
(     

 


 
    /
   *½ ¾+, *¾ ¿+,,
* ½  +     
 
' ½  
'      
(  
   ) * +    #   
  
   ¾
(  
   ) * +   #   
   
    ¾






58

Chapter 2 GRAPH REPRESENTATION



(     

  
    
  
 
   ) *  +
     
' , #
 0 
1 )   
    
 
'  
'
2
0 
1 ) 3 
# 

(

   

  
 
  
 
   ) *  +   


 ,  

 
'    4

 
' , 
    
  
 %  
    


&
     %    
    

 


 



(

     

  
     
 

' , #


 
 

  
' 
0
1 )   
 

 

' 
3

# 





4
 

 
 , 0
1 )  



) *  +  

 
    3 
# 

EXAMPLES



( 4 
  #  &

 
'  &

  

   


 
 


   
   
 
 ! 
 
 "# #
! ! 

Figure 2.1.1

(

 

 
' 
 '   
   # #






)

 * +










3
3

* + *  + * + * +


3
3
3
3


3 

3

3
 

3



FACTS

$(

 &

 
'

  
 
 
 


$(

 ) *  + # % * 

 &

  

  
 
   ) *  + %

* 

5

¾+

 +  


Section 2.1

59

Computer Representations of Graphs

REMARKS

%

(

4
 
    
   
 

 ,  06781, 0

6791, 0:; <31, 0-8=1, 091

%

(

  

,  &

  

  


 #  
  

 
, 
   %  
    

 
     
 

 


%

(

>  
 



 
 ¾

) *

 

 
/


+  ,   &

 
'   &



*

+  
 6#
, #   &

 
', #



  #
  '  
  , #
 #   &

   #
 

 
*

+   4
 
, &

 
'

  
 



#   
 

%

(

?    &

  

   

 
  # &

  (    

'





 
 ,  

    

'









2.1.2 Graph Traversal Algorithms
@      %  
   
       


     
    


 A !
  
!
 


 
/ 

 

 
 /  
 

 



 
  4

  
 / ,  &

  

    
  #
% #

Depth-First Search
ALGORITHM
A !
 
 
      

   
  .  ,  



 
% B#C >  
'    ,   
% BC A !
 
 #
%
 
   # 
'
 

 &
 





, 
%   ,  
  


   
 

  
 # 
 B !
 
C 
  


      
#
 * 
+ 

  %  
# 

  
 


DEFINITIONS
A
 

   

,  !
 
 
    
   

 


    %

  




 
   
 



    , #
 
 
     
  
(



(



(

  

 

 *

 


   *

+ #




 !
 
 
  ' 


+

   
   * + 

  
'    

 

  !
 




(


  


  
  *

 

+ 

  
'



 


 

    !
 




(


   
 
   



 

   




 

  
  

60

Chapter 2 GRAPH REPRESENTATION

&$# '


Algorithm 2.1.1:

#  (  
   ) *  +, #
  )      01    
 
   

 &
  
' 
$ #  ( 

   

      !
 



!( 
 
*+
!  ()  !  !



01 () 
2

!  ()  !  !
 01 ) 
2 
*+2



!( *+
01 () 2



! 
 
'   01 !
 01 ) 
 
*+2



FACTS

$(
$(

 +     
   ) *  +
. #

  !
      
'  #    
  B*C  
     
 
  B+C,   / 
  !
      
 
A !
 
 %

* 

5

'
   #
  
 

 
 

$(

.   !
 
   

 
 , 
    
 
  
 
% 
REMARKS

%

( A !
 
     
  
       
 
=D3 08  681   
 Æ
 
  
    !




%

( A !
 
 
       
 
  
 
 

    


    

 
   !   




    

 
 
Breadth-First Search


!
 
  
  
 /  
' 
  
   . 


   
!   
'  
 #
 


     
  
  

  

 
  . 

 
   
!
 

   
    
'  

 
   , 
      


 
       
   
  
     

Section 2.1

61

Computer Representations of Graphs

ALGORITHM


!
 
   

    !
!
  

   
  


   
 
 !
!
* "+ 
 
'    
%   /   "
  
  
!
!
*"+
     
   /   "

!
 
     

    
 
  
   
 
  
,      

  
  
  
 %  


   

 5  .

,  !
 
        
     

 
 
%

%      

 
Algorithm 2.1.2:

)
&$# '


#  (  
   ) *  +, #
  )    , 01    
   
 

 &
  
' ,      
! 
 
'
$ #  (  
!
 

   
    

  



 

!( #
 
* +
!  ()  !  !





01 () 
2
 
01 () 2

01 () "2




01 ()   
2
 
01 () 32
   " /   "2
 !
!
* "+2
*" "     !
 () 
!
!
*"+2
! 
 
'   01 !
 01 )   
 
01 ()   
2
 
01 ()  
01 5 2

01 () 2
 !
!
* "+2








DEFINITION



"

( ; #$   
 # 
 , 

  
  
01  
, 
 *
01 + 
01  
 #$      



 #
 
* +

"

FACTS

$(


!
 
 %

* 

5

 +     
   ) *  +

62

#
 
* +

 
01

$(



Chapter 2 GRAPH REPRESENTATION

     
  


  

REMARKS

%( ; %  !
 
, 
!
 
     
  =D3 -


 
   
!
 
 
  " 
  
   #
 



 


%(      
!
 
 
     
%   

  
 
  A &%
E  
 
  
  
E

 
!        


2.1.3 All-Pairs Problems
  
 
 
 # 
(  

    
  
#  
  

   

 
    


    
  


   

 
  4
 
  &

 
'   


  
 
 
All-Pairs Shortest-Paths Algorithm

<   #   
      
    # 
   
        # #  
   
 
    # 

 
      
   %#  %#
 >
 

 
  
 

  
   
  # 
  
  
 
  
 
  A &%
E 

ALGORITHM

  
#     4>
 
 #  


 

 
    4>
 
   &

 
'    #

    

 
   ) *  +    

   
  
    
4

   # 
    
' % 0 
1    
   * 
+ .

    % 0 
1,  #   % 0 
1    !  >    

 
  
 4>
 

  
 
 

 , #
 0 
1   

 
    
 
'  
'
 4
 
  #
%



,
   
  
 
   


   
 
FACT

$(  4>
 

  
 
'  
  
#  
  

   

 
   ) *  +  *  ¿+    *  ¾+
 

REMARKS

% ( 4
    
    4>
 
   
 
06781  0:; <31

Section 2.1

63

Computer Representations of Graphs

Algorithm 2.1.3: $"!&+
#
""
# 

(  

 
 

% 0 
1

( : 
'

 


$ # 

 ) *  +, #
 

) 

  



 
 
'

0 1 #
 0 
1  
  
  

!( &'*+
! ()  !  !
!
()  !  !



0 
1 () % 0 
12
! ()  !  !
0  1 () 32
!  ()  !  !
! ()  !  !
!
()  !  !
 0  1 5 0
1 ( 0 
1 
0 
1 () 0  1 5 0
12



%
( ;  0 
1  
  
   
 
'  
'


 
  
   
'  
  
 , '
    
   
>

  
      0 
1 )  * ½   ½0  1 5  ½0
1+ .
 ' 
  #    4>
 
    

  FE



Transitive Closure

. 
 #  &  #  %# #

 '    
 
'
 
'
   

   
   ) *  + >
   


      

      

 
   ) *  + # 
&

 
' , # # 
    
' $ 
  $ 0 
1   

    
 
   

,  3 
#  >
 $  
     &

 
'
 
  

 
 #    
  4>
 
 '
 
      
 
 

    
    
  
  
  
,  
    
 

FACT

$  
%!
*+
*  ¾+  


$ (  


* 

¿+

  


   
  

 

REMARKS

%(  
  

 
     < >
 0>G 1

 

64

Chapter 2 GRAPH REPRESENTATION

Algorithm 2.1.4:
# 

,
#- ."!#(

(  

 
   ) *  +, #   ) 

0 
1

   &

 
'

(  
  

 
' $ 0 1 #

   
 
   

, 3 
# 

$ # 

$ 0 
1    


!( $  
%!
*+ 
! ()  !  !
!
()  !  !
$ 0 
1 () 0 
12
!  ()  !  !
! ()  !  !
!
()  !  !
 0 
1 )
"# 
0 
1 () 0  1!0
12



%(

; $  0 
1 )   
       

 
 
'  
'

    
   
  
'  
  
 , '
  
    >

  
    

%  0 
1 ) %  ½0 
1 ! %  ½0  1
 %  ½0
1
#

  ! 
       

 .  ' 


  #
#    
  

 
    

  FE 


2.1.4 Applications to Pattern Matching

    &

 
 
     
!
   
 


      
 %   
 
    


 
   
   
 

 "   

 
 

 
'
  
    
  '
 


  

 
 


  
  #     
 

    

 
  
 
  
    < : F 




 

 
 
 
  #  

   

 
 
DEFINITIONS

(




    


*?4+

 ) *  +  #


  
'       



    

 
     

  , 

 
 


'

&





 
       
   H  ) #

H   !   

# 
 

, 

)    
       


Section 2.1

65

Computer Representations of Graphs



(  ?4    
 *  
    
! 
' #       *

(
 (

 
  
 
'  

 
   
 !  
. +   
   
,  

 
+     

 *& *     &    

( ;      
£ A!  ) ) 
 )   
 
½  
  

   ,   ,  !  
   
 

    ?4









 













( ; H   !        

      
 !


   #(





H

,  
 
'
       
 )  
 
'
    )
 4

   H,   
 
'
    
 .    

 
'
       +   , 
* 5 +  
 
'
       +  ,
  
 
'
    +  , 
*£ +  
 
'
    +£ 


 

 





>
   #
   
  
 
'
      

    F

  

£   



 

  
5,
 

    



  5 4
' , ***-£ ++ 5 + 
 #
 -£ 5   
 
'
      
 -  3  
Kleene’s Algorithm

< : F
  
 



  
 
'
  
  

  
!      
, # #, 
   4>


   
  

 
   


ALGORITHM

;  ) *  +   ?4  #
  

 
  
     FE

 * #+ #
% 


   / 
  

 %   #
 

 %  0 
1  
 
'
  
  
 
'  
'
#  

  
'    *'
    
   +    

 
 
REMARKS

%(
%(

FE 
  
  0FDG1


 



  FE 
, #

  
  
 %  0 
1          
 
'  
'
# 
 
  
'  
  
 , '
        

%   0  1   


      
 
'  
'
      
  
'  
  
     

*%   0 1+£

        
 
'   
'  "



   #     
   
  
'  
  

    
 %   0
1

      
 
'   
'



66

Chapter 2 GRAPH REPRESENTATION

Algorithm 2.1.5: /"0# "!

#  (  

 
   ) *  +, #
  )   ,    
'
0 
1
$ #  ( 
' % 0 1 #
 % 0 
1  
 
'
  
  
 
 


!( .


*+ 
! ()  !  !
!
()  !  !



% ¼0 
1 () 0 
12
! ()  !  !
% ¼ 0  1 () ) 5 % ¼0  12
!  ()  !  !
! ()  !  !
!
()  !  !
%  0 
1 () %  ½0 
1 5 %  ½0  1  *%  ½0 1+  %  ½0
12
! ()  !  !
!
()  !  !
% 0 
1 () % 0 
12




     
  
'  
  
     , 

 %  ½0 
1  *%  ½0 1+  %  ½0
1+

       # 
 ( 
  , 
    "
 

  ,  
  
#  

  
'  
  
        

%(  4>
 
    

  FE 
 #  


 
  %  0 
1 ()  *%  ½0 
1 %  ½0  1 5 %  ½0
1 .  4
>
 
 # E  
 
 
     
 #    

 
   ,   4>
 
   



 


  *+  

  
%(  
  

 
    

  FE 
 #  


 
  %  0 
1 () %  ½0 
1 5 %  ½0  1  %  ½0
1 #
 5


 !  

 


%( , 6
,  7
 FE 
   
   

 
 06781
%( @   %
   
   
        
!  ?4  '
         ! 
 
'
 
   

 
 
% ( 4
 
   !    
 
'
   
 


  
    0=3, <79G1

Section 2.1

67

Computer Representations of Graphs

References
0=31  I , 
 
!  
  
,  DDJ33  
 

 
    
   
, - K I
; #, . 
, ==3
06781  I , K - 6
,  K A 7,


 
  >, =8

    

06791  I , K - 6
,  K A 7, 


  >, =9
0<79G1  I ,  < ,  K A 7,  
   >, =9G



   

   
 

0:; <31  6 :
, : - ; 
,  ; ,  : < ,
  
   !
 . 
, 33
0-8=1 < -,

" 
 
 :






<

 
, =8=

04G 1  > 4, 
 =8 *<
 +,

# $ D*G+ *=G

+, D

0681 K - 6
   - 
&, -Æ
 
 

    ,
# $ G*G+ *=8+, 8 J89
0K881 A  K, -Æ
 
 

    
 #
%,
$ *+ *=88+, J

%#

0FDG1 < : F, 
     
   !   , 
J3  

 
 , - : - <  K 
:
, 

 7 

, =9D
08 1  - 
&, A  !
 
   

  
, 
* + *=8 +, GJG3
091  - 
&, 
 

 
   
, =9

$ %# 
#

 &
'   
 <
 .

0>G 1 < >
,  
   

,

%# $ =*+ *=G


 

+, J 

68

2.2

Chapter 2 GRAPH REPRESENTATION

THE GRAPH ISOMORPHISM PROBLEM
  
    

  I
    

  ! 
 / 
  


 
  .
 
  
 




 D : ' 




Introduction
  

  
, 
   !,  


  Æ


 
  #
#   
  
 

 . 
'  '$,
Æ  

   '    .  ' 
 *%
+    

 
    *' *¾ ++, 
   *' *    ++
  


  
 
'$
  *' *½¾·´½µ+   '$ 
  

  
  #  

  
 '  


     
 , 
  
       ,   
!
 
  
# 

   
   
 

2.2.1 Variations of the Problem
DEFINITIONS

( #    
  ½ ) *½ ½+  ¾ ) *¾ ¾+ 
 

 , 

¾,  
   ,    , ( ½ ¾ , 
  
 #


 * & ½, 
    *&  ½     
    ,**+,*&+ 
¾ <
  &



   &
  , 
  

 


(
# 



(

     
  # 

 L
 
 ,  
   ,    
  !    .  

 L

 ,  
 

      
   $
 
 
 , #  #     

    
 

#  
  ½ ) *½  ½+  ¾ ) *¾  ¾+ 
 

   

  

  ½,  


 ,*+    

   
   
  , ( ½ ¾, 





(   

     
*  +, #
      
 
   

         ,
   #  

 
½ ) *½  ½+  ¾ ) *¾  ¾+ 
 

 ,  ½
¾,  
  

Section 2.2

, ½ ¾
 ¾

,   
 

  




(



,

(



(



, 
  

 

, 
    


 ½
/

 

    
  


½ ¾ / ½



69

The Graph Isomorphism Problem

*

+ )

/ ¾
*

+    




 ½ , 
 ,


,

* +

, 
 

)

* +


  
 # 
 





  
  

    


 
  / 


¾ 2

   

*+ ) ½  ¾      

 ) *½        +  0 )
!½    !   + 
 

      &
    ! (
)       

  

     
 , 
  # 
 
*


  
   
' 



  

  
 


   
 




 



     

  

FACTS

$

(

# 
 







# 
 
  &

 



1
1  ½  1


    
   
   
'



)








 


, 
 

½

    #  !    
   '  

 
  
 
  #     &

 L
 

$

(


     
  

     / 

 '$

$

(


     

  

     / 

 '$

EXAMPLES



(

-'    
  
 
 #

 
 2  

  

  
  
  
     ,      


 
     
    %2

 ,   % 
  
 


 < ,   #     ,   

 
 *
0F=D1+




(

   



2

½      

)

 

   
 



, 

 
 

3 

  

  
#   &

 
'

 

 > !
 


   
*


*

*

+  
*

+  

+    '

 
  
  

 

+  

 !
 

  


   
' 



 

  
   

   
 

2.2.2 Refinement Technique
 
 /   &
  

 


# 
 

    #
#   
 

<       


   
  # 

  
 / , 
  0:831, 0>8G1, 0>8=1    ,   

 #
 

 Æ
 

 
 " 
 
'$ 
 

     
! * 06781, 0-
<931, 0=81, 0:91,

70

Chapter 2 GRAPH REPRESENTATION

0:91, 0AA;881, 04 931, 04M <
< 91, 06;<>981, 091, 06>81,0F=D1,


(

08=1, 0
91+  #
 
%
N  # 



 

'$

*





 
+

N      
 / 

DEFINITIONS

%



(





 
        3 (  %




 

       

 
 
'
 

  
 
  #  

      

  
  

(



(

 



3





 1 

(





(

4



(



5 

6

 

   
  

  
       

 &


4  %½     4  %
* +

* +

  
   

  3

 1

%½    %
%

) 0

 1   #



, 
    
    





   #   




 


½    




%

 
   
 

 3 ) 0%½    %1    
 
'



+ 



  
'

* + ) 0

*



* +

 


+3

(

 




 



 .   



3 


 , 



%½    %

) 0






  
'    

*    
+

5 

6

 * +

4



   # 



3



0

* & %




1,

 

 

%
5 *

6

  #





    #

$

* + )





(



3
*

 

+ ) 0

%½  %¾     %

 1



    

73
*



+ ) 0


1 )

4* % 
*

+



* %

#






*4
 

, 
Æ
 
    

 
(



"

  

  3


!





 * +

,     


  
!
4
 

 3 , !  

 " 


(



5 &

6

(







+

 

 
  
 
 

   

    .  

  #

 
 

*

,   

,  ½
(

*

+

3
 ¾


*

+ 




   



  ½       


  ¾
FACTS

$

(

;








!

+ 3½
*

(

;













*




,

,
 
 ,  

 

+ 3¾
5 ½*
*  ½  
6

+ 



$






  
 



  
  #

,

 



 



    

+ 4


,

*

+

 *






  
 



  
  #





+ )

5 ¾ ,*

6



, 

 *


 3½

* ++

,
 
  . 
  



*

+ )

 3¾
*

+ 

7 3½
*

+ )

7 3¾
*

+

Section 2.2

71

The Graph Isomorphism Problem

$(

; 2½  2   
 

  
    ,
 
  
  
 

    
  



  
  # 
 
  

 2   2 



REMARK

%(

 
 "    
 / 
   
      
  



 
  
   * 0 =31+ 4
   
' 
  , 
  #
   
     
'

8*+ ) 08 1



0 12






0   1

#
 8     
   
 
   
'   
.        ,  
'     
   
  
A
   E % 0A3 1 
, B
       
'
 #  !   
  


      
!
 
  

   
      

'C
Backtracking


! 
 /          
%

%  



DEFINITIONS

(


   
%

%  
   #  / 
   9 2  9, #

 9 
 

  
 
  
     , 
#
 




  
     


9

(

 

   
   #    92  9, #
 9  9 
 
  "    
 

  
 
  
 



( 4


   9 2  9,    
 


  
 
 
',
  
,    


  9 ,   
' 



  


  9,  %   




    

 *&(  +   #

    "    #



   
   
  .     "   



,
 
 
 






( ; %    


  

 9,  * % ,   :  
  "   

   
 9 
 
 % #  #


( *
 %  * >   !'  *   % ,  



 % ¼  : #
  
   %   
  , 
 ( % ¼ ( % ; 
FACTS



% 
  '$   
%

%  
# 
!    Æ

     
  



   
  # # 
 
     '    Æ

   
%

%  
   # 
! 
     


$( 
*0 891,
0 8=1+ .
     #
&
  
   


*

+  




   
E 
  

 
  

%

72

Chapter 2 GRAPH REPRESENTATION

$( *0
DD1+ ; O  
 
     ! 
&
   P,   


  O  P    ,
 
    
¾  , 
¾ 5   
 Æ

  

      # !'   
'    " 

  

    


  





$ ( . 

     
%

%  
, !'   
' 


  


,       
%

%  
  * +,
#
    
'  
  
   /  ,  
     

  *
 +
 '    

 
  #  
 
 , !'   
'

 


  

 
 
  '       
      

 * 0:91, 091+

2.2.3 Practical Graph Isomorphism
FACT

$
( -
   # 
  
       
 


   (


 
 * - J  06781+2
 

  * 6
  > 06>812   4 048G1+2
 
 
  * ; %
  0; 8=1+2




  * 6 06=D1+2
D 
  #      *  
0 931,  4   
04 931,
  
 0
331+2
G 
  #    
' 
 * ; % 0; 9 1+2
8 
  # &

 

         
 *
 , 

,    0
9 1+2 
9 
     
#  * 
0991+
REMARKS

%( -Æ
      >  
; 
 * 
 "  
! +
     0=81

%( 


 
  
   

 
   

 
(, 
   
 A 
F 7  
(, 

    

 
 
  
   

  

 
 2 
(
    



 

     
 

%( 

 
(  #
   
    : 
   
 
7 'L;  ' 
%(  #   

 
( #  
  
 
   
 



 


%(  
%
    
  0
91,    
 
    #
   !
 
  /  

 
( 

Section 2.2

73

The Graph Isomorphism Problem

  
%

%  


2     
%

%  


     

 *'
  

   
, 
#
  
  '

 
+ 
 

     
  , #   
  


    
  
   

 
 

9



%(


(

  "   
  
 
 
  


 ½   )
 8GD,  ( "&) *  !,  #

     "   
 


    ) G888  >   


, Æ
   ,
   
       !   


 
, %   

% #
        


   
 




(

2.2.4 Group-Theoretic Approach
 
 



 * 0>;G91, 0>8=1, 08=1, 04 6; 931, 0; 9 1,
0; 91+

 '$  

 
 
, 
DEFINITIONS



(  

 


  
 *
 +   !    



  
 



  
     
   
  
  


  
FACTS

   
 



    '$  4
      Æ


        

  08=1  0; 9 1 #  


  
 ,  * + 
    



   
    
    
  

 

     
  




 
  #   


,   

%   

    ; % 0; 9 1 

  #    
     Q

; %E 
 / , #      
 

 

 

! 

$





( * 08=1+    
' 
  , !       
'

 *
+  , 



 
      #
*
+    ,
'$  /   
  !     

   * +,    

 
  


 H 
     

* &



  ;



 

$(

! *+   

* &
! 


   
    
 

  !'

*; %09 1+ . 
     
        



  
  * +


     
 

   

 
 

 ,  
     
  
   
   
 
#  # 
 

 
 

! 




  4


 
 

    

 *+,   *

  < 3+
     #   

            

 
 * 

  < 3, ! * +   
   
   
 !'
+
$( 
 '    
 < 3 
  ! *  + ) ! * +  
 
! * + ) ! *+

74

Chapter 2 GRAPH REPRESENTATION

  > 
  

  
< , 


     
 
3 ! * ·½+   

      
   ! * + 


 
 

! * ·½ + ! * +    9 








$













( 
     
  !    

 
 %
 
 
  9 , ,  
   
   ! * ·½ +  !' 


   
   

$(


     
  !    

 
  


   ! * + 
  '   
 


9 *! * ·½ +  
 ! * ·½ +






$(



06;<>981 
  

 

*¿  +

  
 

'$

 



.
 ,  #    '   
 /    ,    
'  , 
 


  
  
 
REMARKS

%(
% (


  4
   4
  
   
  ; %E 
 / 

 
    
 

 

     4
 G  
 %#     

%


(   

  <@:E9 *0; 91+
   
 
   

 


 

  
  *' *½¾·´½µ +  

2.2.5 Complexity
  # 
  

    %#   ?
 , 
 
  . # 
 ;
E 
 0;8D1   1 ) 41 ,  

41 ) 41  1 
 #  
 
 '      '$  


  
 
 
     * 0K8=1, 0FM=1+
DEFINITION



(  

 
 
  
     / 
  '$
FACTS

 

 
      '$
   
 

 

 
 * 0678, =, 89, ::89, 8=, 6<9 , ; 91+

$(

    
 
 
 
   # (
  
  
  
 2

 

  
 2

  
  
  *  # 
 , 
     

   
    
+2

     
 

  # #   
 2

Section 2.2

75

The Graph Isomorphism Problem

D  
  # 


  *   
 

  , !*+, 
#
 ,*+ ) +2
G  
  
  *
   
  

   
' +, 
  
!*++2

8  
  

 *
     

 
9 
       
   

 
    
 2

!*++2

  

 

     

= 
 

 



  
 2
3  
 /  
 *  # 
 #   
   

 
 , 
 , 
, 
   
, 
    #

 
 / +

$(  41  * 
  
 




% #
  
   , ( ½ ¾   
 +
$ ( *F    %% 0F3 1+ 
    '  
 
  
   %'$,   # 
%
 
 
REMARKS

%( @  
 

'$  ' 
    
 
 '  
  


   %'$,  
  ? 
  
,   $
 

  
  
 %#   )
  4
 = 

 
 
  
 
  

      '$

%( .  #      # 
%
 , 

   
 ; <
%
*0<8 1, 0<8G1+,  
 

References
06781  I , K - 6
,  K A 7,
    

 
,  >, =8
08=1 ;  , B:
 
  
  
   C,  
,
=8=
0-
<931 ;  ,  -
M,  <  <%,  
  
 ,  $
%#  # = *=93+, G 9JGD
0
9 1 ;  , A R 

,  A   , .
   
 
#      
,  # +,
# $ #
   # *=9 +,
3J 
0; 91 ;   -  ; %, :
    
 ,  # +-
# $
#
   # *=9+, 8J9
0=81 ; , . I : ,  F , A I 
 %, 

: 


 
 : 
   
 .. 

 .  
  >  
; 
, 7=83, K 
=8, 
 
 7 
 M
M


76

Chapter 2 GRAPH REPRESENTATION

0=1 A  ,  
 /  
  /   
  
 ,  
    .

 D *==+, GJGG
0991 6 ; 
,   
 



'  
  

   
  
, %#  
  *==3+, GJG
0891 F < , .
    

 ,  
   !   

   / ,  $ %# # 8 *=89+, 8J 8=
0
DD1 6 

%, A $

    !  
 , # 
 # $
# # 89
*=DD+, GJ9
0::891  K :
  : K :
, 
  
   
 

 
 ,  " & ' 3 *=89+, DJ =
0:91  K :
  F < , ; 
   
  
 

, 
 
 ,  

 ,  $ %# # 3 *=9+, 3J D
0:831 A  :
   : :  ,  Æ
 
 

  
 ,
%# $ 8 *=83+, DJG
0:91 A :
    
,  

 
 


  

   
 , %#  
 D *=9+, DJG 
0AA;881 ? A, K  A ,   - ;
,  # 
 
 
  

 , ( 8 *=88+, GJ3
0A3 1   A
   ; < E %, 

    
   


', " 

  &
  & ' *  S;... * 33 +, ?# R
% 
 <
,
9JDG
04 931 . < 4   K ? 
,    
 

   

   
   !'  ,  # +/
# $ #
   #
8 *=93+, GJ 
048G1  4, ; 

 
  
   

 ,  J
    # 0  # 

 .      *< 
 
  
, -+, - 
 7  
, =8G
04M<
< 91  4M

, > <

,  - < 
%
, ?
 
 

 
 
 
     
,  # +-
# $ #
   #
*=9+, GJ83
04 6; 931  4
, K 6
,  - ; %,   '   
 


 
  
 , 

  < 
 :  : 

, 
 

,  :  *=93+,   & 
  *=93+, GJ
04 6; 931  4
, K 6
,  - ; %,     
 


   
 ,  # /+
!!! #  1
 # # *=93+, GJ

Section 2.2

77

The Graph Isomorphism Problem

06;<>981 T  , :  6$, -  ; %, :  <


,  >
,
 *¿  + 
  
 *¿+
  

   



 , %# $  *=98+, DJD
0K8=1   
  A < K, 



   &
 , 4
, =8=

 


  "


091  
,  

 
 
 
  
 
,

  $
# G *=9+, =J G
0
331  
, .
    
 
  
  !  ,
 # 0/ $ #
   # * 333+, =J G
0
R=91 K ; 
  K R,
===

" 

  
 
, : : 
,

069 1 4 6

,  ,   <,  
  
 
 
  /       
%
 

 

 ,   
%# $
#  *=9 +, DJ 
0681 K - 6
  : F >, ; 
  
 

   

 ,  # 2
# $ #
   # *=8+, 8 J9
069 1 :  6$, 
 


  
  .
 , . 
 
&
  
  G, <

, =9 
06=D1 > 6 , *?+ 
 


    
 
 




 ,  $ %# #  *==D+, J=
0F=D1  F , : M
%
,  M
%
,   
, 

: 

  

 : 
   
 . 
   
   :

*: 
+ 
 7=D3, A"
=D, 
 
 7 
 M
 M

0F3 1  F   A  %%, 
   
    '  
 "
       


 ,  $ %# 
 
* 33 +, D3JD G
0FM=1 K FM
, 7 <
M
 ,  K 
U,


   
, 
%M 
, ==


" 
 
   


0F
=1 A F

  ;  6 
, @ 
 '   
 



 ,
$
# 

  8 *==+, D8J 8
0F
3 1 . F
 %,  ;,   A , 7 
     
 

   
 , 
$
# DG * 33 +, 9=J=
0;8D1  - ;
, @  


     

  ,

# $
# *=8D+, DDJ8

%# # 

0;==1 K  ; #, 
  
, : 
3 *  D DJG+ 


 
  , - 
, ==

 

78

Chapter 2 GRAPH REPRESENTATION

0; 91  ;  #, < ?
 
   
 
  
 ,  $ %#

 3 *=9+, J 
0; 8=1  < ; %
 F < ,   
  
 

  
 
 

 , %# $ G *=8=+, 9J=D
0; 9 1 -  ; %, .
   
     

    
   , %# 
# 
 # D *=9 +,  JGD
09 1  !, 
  # 
  
 '    
   
%  
 
, 3
# %# $
# 4  5/6  *=9 +,
DJ8
08=1  ,     
  
 
  
,  

   .

 9 *=8=+, J 
0
91  A 
F, 


 
  
 ,   & 
 3 *=9+,
DJ98
0<8 1  
 ; <
%
,  / 

 

 
'
 
#  / 

/
 '    
,  # 
+0
!!!  
'

  


  *=8 +,  DJ =
0 891,   
, @  
 
  
 / ( 
  

 
,  # +7

# $ #
   
 *=89+, DJD9
0 8=1,   
, 
  
 , 


%, %# 
 
 #
*=8=+,  9J 
0 931,   
, .
    

      ,  # +/
#
$ #
   
 8 *=93+, DJ D
0 =31,   U
, A   
 #  

 
 ,  88JD  
 
  
  $   
 *  K     
,
-+, > , ==3
0<8G1 ; <
%
,      

,

 
  
*=8G+, J 
0>G91,  >  
   ;, 
    
   

 
  

 
  
  

 , &
 

  
 5 # /6
= *=G9+,  JG
0>8=1,  >  
, *+ @


    !
   
 , . 
 &

 $

 DD9, <

, =8G
078G1 K  7,  
 
 
  
 , %# $  *=8G+,
J 

Section 2.3

2.3

79

The Reconstruction Problem

THE RECONSTRUCTION PROBLEM
 

 
  


 # 


  :&


 < 


  

  :
 


  
  
  A
%
  E  F
E  
D ;U"E 2 ?>  E ;
G A 
 
8 .   A
%




Introduction
.  !
        # "  *=88+,  &
  





"  


  :&

  B 



 

 
  
C   D 
  , # 
   #
 

 
     
 >   
 
   

 #  
 
 %#   


  


     
 , 
  
       

2.3.1 Two Reconstruction Conjectures
<


  
 
   #   (   

 
 #     


  ,   ¼ 
    / V . 
 

 # % # %#, 
    


 ,   Æ
  
 
 
 
  > %#, 

,   



  
  

 
     %#  
       
  

  


  




¼

Decks and Edge-Decks
DEFINITIONS

( ;    
    

 4
 
'   ,   

       
      
   4
 
,
    
  
,       



(     
  ,  *+,  

    
'  

     ,  *+,  

      
 
 
 
    
% 
  . 
  


'  

 ,  
  
  

   *+    
 
%

80

Chapter 2 GRAPH REPRESENTATION



,  
%   
% 
   ,

 ,  
 
   
 

  .       

   
  ,       
 
  
      

    
  
     
 . #        ,    #    
     
       # 
  
 .  
   
  
   A!    


,  &
 

(% 





  


% 

EXAMPLE

(

4 
  #  '    
    
%

Figure 2.3.1

 

 # 1

Reconstructibility

#  
 
 06881, 0?891    
   %#   



  
, 
#  
    
 
  < 
 ,   
 
 
'  
 

     0-991, 0991, 0;981, 0=1 

  0;<
31
  

 
  



 

DEFINITIONS



(  
  0 #    
%   
  


  . 




     

         





(   


    
  #
    
%  ,
  
         

  
 



   
 

 
CONJECTURES

<     0 
 
  
  *0 + ) *+ @



  #
   

  >     



 
 
% 


, %!#(! .!( 0FD8, 7G31(

0

* # +    %

    

, &%!#(! .!( 06G1(
 %  

* #      

.

,    
% *+  

, 

 
 
    




% #
 
      
% 



    /     
 / ,  , #
  
 #  !   
  #     
%

Section 2.3

The Reconstruction Problem

81

EXAMPLES



( 
  
  .¾ 



 , 
   
  .½
   
#  

  



   ,   

  


  
 


  :&

, .¾  .½ 
   



  
 



(  
   ) .¿  .½   



  
 ,   
  .½
¿ 
*  +.½   



   *+    

  , .¾ 
 



  
   0 ) 1¿  .½ *#
 1¿     
 

+,
 *+ ) *0 +,     0  


    - 


  :&

,
 
   
   
  



 
Relationship between Reconstruction and Edge-Reconstruction

.           




  
  
  
%
 
  
%2 
 
 
 
 
    
%,  
 
  
 
 
 
  %  
  
   
'  

  ., 4
        


  :&

  
 

 
#    

,     - 


  :&

 4
 #
 
  



     

 



 
FACTS

$

( *
#E 
+ 0
81 ;    
  #    

 

%    



 ,  , *+   /  
  
 *+


,   



 ,     



 

$

( *6 
E 
+ 06G=1  
   



      
  
  



     .¿ 
REMARKS

%(

> 
   

,  
#  ,   
   



    



%(

 - 


  
   #
 
   

, 
  

  %#  



      
 


'



 ,   
 
 /      




  >
 
 
 /        

 
Reconstruction and Graph Symmetries

  
    Æ
 



   
   
  
%   

      
    
% < , 
 
 ,   

   

          


  
 
'  
 
,

,  
  
   / 



 
  
  
  
 
  
      
 ,  

      
#   
  
      
 
  
   




    
      



   
DEFINITION

( ;    !'   
 

 
   
        

 , #
  # 
   
   

  ,  
        #

82

Chapter 2 GRAPH REPRESENTATION


  

 . 
#
, 

   $
  











,   #  
  





 
  



FACTS

$

(

.




 


·½

,   
 




,    
 




, 

  
  
  
 

$

(

0F8, M 8G, =31 4
 !'



,  
 
  
 


 
 
    
  





$

(

 







,  



    
 
   





099, =31 .  
 


 


¿

,  
 





 / 


  
  
    
%

REMARK

%

(

  !      

!  !
! !

4
 D
      


  
 ;



  
  
  
  
% 
 

 !


, 
 




¿

! 


!
!




   #
 # 

!





 
 


 /  
  


  
    # 
 

  #
 # 
 

< 


, #


   

 #
  

, 
  



   
'





 / ,  #         


 
  # 
 


% 





!



  !   


 &      
 

 &
 



#


  

   
 
 
   

 



2.3.2 Reconstructible Parameters and Classes
Reconstructible Parameters
DEFINITION



(











 $
  

%     



 , 
 
 

 






  

 
;

0





 
 



# 

 
)

, 



  *




 +



 # 
  

0

 





   
  
  

FACTS

     

 
  



  
   ,-



,

4
 

$

(

  
 

    
  
 



  





  * 0;<
31+

$

(

 


/ 






 

 



 

*< 0=,

;<
31+

$



(

   
 




  
% 



,  
 

    
 




  * 0;<
31+



     
'

Section 2.3

The Reconstruction Problem

83

$

( 4

  #    

,  

 



    



 ,  
#E 


$


(    
   
  
%  ,  
     

 # 
#
     
 
  



 

$(

*FE ;+
0FD81 ;   0  
  #     
 


 



  
 *+ <  
,       
 0  





  
 *+


  0  

 
   0 , 
Reconstructible Classes

> #   
   
   B



 C,       
% 
 
%   
 ,    
    
   




    .   !
  !
 
  
  
%    
     , 

,      
 
  
 ,   



  
#    
 '
 !     # 
DEFINITIONS



( 
   
       
 " *
 
 "+  


    , 



  *




 +        -/ ,
 

 " *


 "+  
  
  
 *+ *
 *++
#
    

(

 

      

 *
  




  *




 +      

   

  -/ ,   #%



  *
#%




 + , #   '
 
       , 
  
 
 /  
  
% *

%+
 
 



+ 

FACTS

 #

 #    
 



     
   
# 
 * 
  



  
  #    

 




  
 



 ,  
#E 
+

$(
$(
$(
$(

 

  




  * 0;<
31+
A 

 
  




  * 0;<
31+
0FD81 
 




 

0G=1 < 
 
  * , 
  # 

   + #   


 
  




 

$(

0R991 



 
&

  
   

 
  





 

$(

0
881 
 

  #   
     
#








 

$(
$ (

04 89, 4 ;9, ;91 '  

  




 
0 8G1 @ 


  




 

84

Chapter 2 GRAPH REPRESENTATION

$
( 0
91 .                 


  

 

 
  
   E 

,   



  .



,    
  
    ,   



 

    
 



 

$( 0R 9 1 . 
 '    
 

   E 

, 

 #  

 



$( 04=1 

  #      
    
 



 
$( 0T=9, T=91  
       
   
   

 





   
   
   

 




  3
 



   
    
   

 H 




=*H+
 



    *+  =*H+

$( 04> >31 <
 
 #
% * , 

 
  #   
     

. + 
 




 

$( 0:81 .  
  
 


   




 

$( 00-S 991 :#
 
  
 



 
$( 0-6981  
 
  
 



 
REMARKS

%(   
       
  
  

 .
   
       ?>  E ;, #
 #   
 #
%(     
 
  # 


     #  


 * 
   

   
,  



   

+  '  #   



   

 
  * ,  

 
 
 

 ,  



   + 6#
,  

 , ,  ,    '
  Æ
  
% 0<
91

%( 06881   
 



    



 
 
  
  ?

   
   

%( 
  
   
    
  
  
!


 
  
  

 
 ,    

 
 / 
  
 

   

 
   

,   #

 
     
     D
 
 

 , 

 


  



  
 

2.3.3 Reconstructing from a Partial Deck

    
    
  
      
%


% . 


, 
   #  



    
 


 *

  
 + 06GG1, 
 


 


*

  '    
 
 

  
+ 0G=1

Section 2.3

85

The Reconstruction Problem

Endvertex-Reconstruction
DEFINITION

(       
    

   
     
 


 #  
     
    
'



      /  
 
  
'
%

FACTS

$(
$ (

06GG1 
 
 
'



 

0
81 4

    
, 
 '   
  # 
  
'



 

 

 


    D 
  
 
   

   
   # 

   
'



   
Reconstruction Numbers

       
    
% 
    




 ,
6

   069D1 

  !   



   

DEFINITIONS




(  



   
  ,  
*+,   
 
  
    
%     
     /  
 
  



,   
*+,    !



( ;   
  
    



   
  
  ,   
*+,        
  
    
%   ,

#   
       ,  
     /  
 
   



,    
*+,    !
FACTS

4
       
   

 
  #   

  


  



   
/    5   
   

/         #    5  




   



 
 

$
(
$(

099, =31  
 
  



   
/   

0=31  

 
  # 
    






   
 . 
  
 

   

 
,  



   

  /    5 

$(


$(
$(

0;3 1 . 



   
  

 
      5 

  
   . 



06981 .

  
   
       ,  
*+ /

0=31 



   
 
  

 

86

Chapter 2 GRAPH REPRESENTATION

$( 06;981 .   
  '  

     '  



*+   

  '  

  # 




   
/    




"

$(  
 
   



   
/  

* 0;<
31+

   

 
  . 
   
 
 ,
 


 ,  
*+     ., 


,     


  
  
 .¿  .½
¿,  
*+     . 

    
  


   ,   



 
 

       5 
$( 0=D1 ;

$( 0=1 -
 


$ #       
*$ +  

CONJECTURE

6

  ;
06;991( .
 

  
  
     
,  
*$ +  

FURTHER REMARKS

%(   
  
  



      





 ,  

    '  # 
*+  
*+ . 
,
 



   

 
 
   

 



 
 


% ( >    
%  
 




 ,     




  

 
,    
 



  


   
 
      

 
  
 , #  
  # 
 
 ¿ 




   
,   0991 
 

  
 
  
  

  #   




   
 



   

 

     %#

%
( 
 0991
 



   
      
,

       
 
  
  
% #
  

  
   ) 
   
 
   
   /  <  ! 
     
    
   
  
  
 
#
 
    

 ) 
  )
 
   
   / 
@ 
 
        





   
, #

  
  Æ
  
%   *+



   

Set Reconstruction

6

 06G1   
# 



 ,   < 



 :
&


CONJECTURE

 #  +        %
  .  
    
#  #   
. 
#
,   #     
  
 
 
 
  

' %!#(! .!((


%,     %# #    
   
   
   
%

Section 2.3

The Reconstruction Problem

87

DEFINITION



(  
  
 


  
  
 
 
    


 
        


FACTS

$ (

08G1   
      
   
   




 

$
(

08G1 4

 
   #
  
'      
     
,
 
 / 
  



 

$

( 08G1  
 / 
   
  #      
     



 

$(
$(
$(

08G1 

     
   



 
08G1 ?

 
  
 



 

< 
 
  * , 
  # 

   + #   

  

 
 



 

$(
$(
$(

0831 
 
 



 
0 8G1 @ 


  
 



 

0
:81 7



  * , 
     

+ 
 




 
Set Edge-Reconstructibility

   



 
       



 ,  , 

   
 
      
%    >  




  
   



  #       

 >
    #
    



   
FACTS

$(
$ (

08G1  
 / 
   
    



 

0A4 3 1  
 / 
   
  #    
   / 

      
 / 
     
  #   #


  '
   

,       
 / 
  
  
  
   
 / 
   
 

$


( 0A I=G1 .  
   #    
    #  


  
 ,     



 
Reconstruction from the Characteristic Polynomial Deck

<
#% 0<
8=1
  
 



  
 





   
  
    
%, #
 #
  

  6  
 #%   
   # 
/
  



   





    
    
%  
     ,  #

    
 
  

  
  #     

88

Chapter 2 GRAPH REPRESENTATION

FACTS

$

(

0<
8=1 




    
  



 

 


 
    
%

$

(



.   
    
% 

 




  # 
 

,  




  







  
   


% * 0;<
31+

$

(

0:;=91 




    
 



  
  

  
%

$

(

0<
1 .  
   




  

;

 

  
   






  



  
    
%

Reconstructing from -Vertex-Deleted Subgraphs





'  
  



   
    




  



 



    
   >   
        



   <
  D # #
 



  
  6
 #   


  
 

FACT

$

(

09=1 ;



   
   
 / 
    Æ
 



  
 
   
 
 
 
    
 








'  
 
* + 

, #


. 


,  
 



* +  

   
   



2.3.4 Tutte’s and Kocay’s Results
. FE ;  
     
  



,    #   




 
   0 8=1 # # FE ;
  ' 


 
      
 , #
   
  

/ 
 .
0F91, F


  E
  #   



F
E 

 

 
'     0=1  0;<
31

Kocay’s Parameter
DEFINITION





(

;



  
  


   

+

 
  
* +

 
)









) *






½  ¾    
 


+  / 
  
  * $


* 

   
+ 
 (

    


 






½ ¾    
        
  

  / 




* +

) *



  

 + 

) 



*

+



Section 2.3

89

The Reconstruction Problem

FACTS 0 8=1, 0F91 *  0=1, 0;<
3 1+

$(

;    
    ) *½       +   /
 * (  *+  ; >*  +   

! 
*  + 












  
  #  




#
      % 
 
   
* + ) * +   * + 



 





> 

 
  
 



 # 
  
     !    

 
  

$(
$(
$ (
$
(

 



 

    
   



 

 6  

   



 

 

     
    #    
! 

  
 

 
 
 

 
 



 




The Characteristic and the Chromatic Polynomials
DEFINITION



(       
   #
 

    
 







  4
 
 
  




, * +    

  



 * +

FACTS

$(

0<G1 *  0 =1, =+ ; 




  



8

5

8



8

5 



5

 

Æ
     

 )

*+   

5





#
     ' 
 
   




$(

 

 
 
  

0> 1 *  0 =1, 88+ ; 


  



 

- * 5 - * 5    5 - *
-

 

Æ
     

- )




*+  




#
     ' 
 
   


 

 
  

 

 

90

Chapter 2 GRAPH REPRESENTATION

$(

0 8=, F91 




  



  *<  0=,
;<
3 1+

$(

0 8=, F91 


  



  *< 0=, ;<
3 1+

2.3.5

Lovász’s Method; Nash-Williams’s Lemma

.  /  
        

    




  
  
    
  ;U
" 0;8 1 #    
 
     
    
 ,    



 ,
%   

   
    
  '
  

 7  
   ;U", M
0M881    



   .   / 

  
*       

  ;U
"+, ?>   0?891

   
 #
 ;U"E  M
E
  #  
 '
'   
     0-991 
0=1 
0;<
31
The Nash-Williams Lemma
DEFINITIONS



0

 0

( 4
   
    ,  


  
 

  ! 
  &
   

' , 
       
   
  
         

0



    
 
 
   0    00 1

(



0

   
  , #
 

       
   * +

            ,        
  


    &
 
 

  





   


*+,    &
   

'  

0
0

    
 
 
 
  0 1 

0



 0 #  
  



REMARK

%(

0

0
0

?
  0 1   0 1 ,    
  
  

   
   0 1 ,  
     
  
     6#
,  
   0 1  , 
   
 
  
  



0

0

0

0

FACTS

$(

0;8 1 ;

  0  
    *+ 
0

0 1 )



 

* +  0 1

0

 

Section 2.3

$(

91

The Reconstruction Problem

0?891 ;   0  
  
  0    
% 
00 1 )  *+ 5 * +  *00 1 01 +

)%/
-

  









*+, 



*< 0=1, 0;<
31+

$(

0 , 

*:

  ?>  E ;+ *< 0=1, 0;<
31+ <  

   ?>  E ;,     ,

* + 
* + 

$(

 0

0 < 32
 ,  01 < 32
 
1 ;    
  
   *+ < ¾ ;
 ,  0 1

0;8




 

$ (



0M881 ;   
  
 




 

 

 

,

 

< W   

$
(

0=31  6   
  #    Æ
 
  
 

 




 
  
   
    
%      
Structures Other Than Graphs

     
     , # !  

    

* X + #

  !  , X   
  
   
   , 
       



    


, #     


  ,  Y
 E   
 X  /    #

 



  / ,    
    
 X

  
 *









 





. 



  

 ,
#           


,     
' 
   



,  X    



 #   
 
    

,   
 
  







DEFINITION





( . 
       ,         

 ; 
 
  



 
  






FACTS

> #
 
      #  



     


    !

 ,    
 
#
  
    ?
>   ;  


,   %  #      
 
  



/
   
 
'



     ' 

 



         
  0=, ;<
31

$(

0



0;<
31 ;
  
  #      
 ,      

   

 
   #  
  
 


'



    


 * +

0



< 0;
$( 0:F
9=1 ; * X +   





  



 

<

 
 

0

  



<

X   

92

Chapter 2 GRAPH REPRESENTATION


 $  <
 0 <
=91
 




       




? 
 
, +,  
  
 

           ",
  
 

$(

0 <
=91 <    
  
   

 

     

$(

0 <
=91 4

 

 ,     


   
 

$(

#


$(

?

 



 

?






  
 

0 <
=91 4
 , 
    ? 



  
  =@*+ ,
* +    
   

 

  

@ 

0 
==1  
 !     + * ,    #

   ! 
 
     !     "   + 



  
 
 

$(

0 3 1 : 
# 
    +¾   

 
  


   
  
  

        =3 
 

 !        +¾   /  
     D    


     ,    
 2  ,   
   
 



 ,     



   
D
  
0:F
9=1      

    
  ' 




   


   

   



 
The Reconstruction Index of Groups

;%  



   


 * X  +     
  
  

 

   
   
 X
DEFINITION



(  


  :*X +   
   
 X 
   
    
 
  
   
 #    ,  



* X  +  



 
FACTS

$(

 - 


  :&


 
 
(  A )       , 
    


,   
 
  A ,       


  A


    
,  :*  + ) 

$

( 0=91 



  '     
    



 
'     
   D
4
  
   



  '  
 ,  0:=G, =G, 98,
= , =D1

2.3.6 Digraphs




 
&

 
 
    

   
 



   
   

   
#  



  , # 

Section 2.3

93

The Reconstruction Problem


  ,       
    
   
 
   
' 

  

DEFINITION



( 0 =81   
        4 

     /  
 
  
 *   
 * +   * ++, 
 

   
FACTS

$
(

0<88, F9D1 
 '   !     
  
 




 

$

( 06G81 
    ! 

  
  








  *<  06881+
<      
   



     

   
 0A =3,  =G, I ==1
CONJECTURE

, 4 &%!#(! .!( ! 
# * 

+(


4

* #

%  

2.3.7 Illegitimate Decks
DEFINITIONS


(



   
 

      


 



 

      

   
   
       

   
%
(    
   
  #

   

 

 
     
%   
 
FACTS

$(

09 1 A
   #
  

   
       
%
    
   
 


$

( 06<9 1  
  
 
    /   
   
%
 

 

 

 
    
  # 
  
 '   
    
%
   
  
 

    
0FM
<
=, F
6=1

References
0:F
9=1 ? , R :
, . F
 %,  R  , : 





 
, %# #
  5 # (6 8 *=9=+, DJG

94

Chapter 2 GRAPH REPRESENTATION

0A I=G1 ; A 
, < A ,   A I

, @   




 
&

, %# # $
# # 
# 3 *==G+, J=
0
:81 - 
&  A :
 , 7



    6

E :&

,
# $
# (# 8 *=8+, D=JD=G
0;3 1 F K 
%  K ;
, @  

 
  #  




 
 
,   # G * 33 +, 8J9
06981 A > ,  - 
% %,  ; 6 6, :



  
 
 , %# " 

   *=98+, J 3
0 =1 ? ;  ,    " 

 , :
 7 
  
, ==
0=31  U
,  
 
  



   
, %# " 

 
 *==3+, J
0G=1 K  , @ FE


 
 

  # # 
# #
GD *=G=+, 98J=8
0G=1 K  , @ 7E
&

 
 
 
 , 8 %# $
# 
*=G=+, 9J 99
0=1 K  ,  
 




E  ,  J D   A F#
*-+,  9   
 , :
 7 
  
, ==
06881 K     ; 6 
, 
 



  N  
, %#
" 

   *=88+, 8J G9
0
81   
, @ 
&




  



   
 , %# 
#
   *=8+, =J
0:=G1  K :
, <
 
   



   # &#  *==G+,
J
0:81  T : ,  
  #          
 


 
  



 ,  8J8   :  
, K  4

   F
 % *-+,
:  
  " 

 , <

I
, =8
0:;=91 A  :% U
  ;  U
, <% 
 
'   




 
&

 





   
      
 , (#
# # $
# &
# # $
#  *==9+, =J33
0A4 3 1 : A
, @ 4
,  A  
, @ 



   

 / 
, 
$
# D= * 33 +, =J33
0A =31 A : A
  :  , @ 



   
 
,
%# #  # 
 # D *==3+, 3J 
0-991  ? - , 


  



 ,  # & # G
*=99+, J 3

Section 2.3

The Reconstruction Problem

95

0-S 991  ? - , ; 
 R S ' , :#
 
  
 



 , %# " 

   *=99+, DJD
04=1 6 4, -



   

  #      
  

N.I, 
 # $
# # 8 *==+, 9J

04> >31 6 4, R; > ,  : F >, @ !'    




   
 
 #
%, " 
 # 8* + * 33+, J D
04 891 < 4 
   ,  
  

  #    
  




 
, .., %# # # # # *+ *=89+, 33J G
04 ;91 < 4 
  K ;
, 



   '  

 , .( 

  , %# #
  5 # (6 3 *=9+, 99J=D
0 8G1 >   ,       


%, %# #
  5 # (6
3 *=8G+, 3JG
0
91 : A     A 
F, < 


   




    
 
 , %# #
  5 # (6 3 *=9+, 9DJ 9=
0
81 A ; 
#, 


  
 ,  #  # $
# # 3 *=8+,
J
0 =G1 :  ,  


 



  
, 
$
# D
*==G+, 8J9
06G1 4 6

, @ 



    
  
 

    
 , 
8JD   4 
*-+,
   " 
  
 
, 

 <  
<
 =G, 


, =G
06;981 4 6

  K ;
, 




   
  '  

 , " 
  
   *=98+, DJD
06;991 4 6

  K ;
, @ 




   
 
, 3
#
%# $
# 4  5/6 = *=99+, 8JG3
06GG1 4 6

  -  
, 



    
 
  ' 
 
, # %# $
# 9 *=GG+, 93J93
06G81 4 6

  -  
, @ 
 



   


  
, $

# $
# 8 *=G8+, J 
069D1 4 6

   ,  
 



   
, %# " 


  = *=9D+, DJD
06<9 1 4 6

,  ,   <,  
  
 

   /       
%
 

 

 ,  
  %# $
# *+ *=9 +, DJ 
06G=1  ; 6 
, @



   
 ,  #  # $
# #
*=G=+, 9DJ98

3

96

Chapter 2 GRAPH REPRESENTATION

0FD81  K F, 


 
 

, 8 %# $
# 8 *=D8+, =GJ=G9
0FM<
=1 K FM
, 7 <
M
 ,  K 
U,
" 
 
   
 

   
, 
%M 
, ==
0F91 > ; F
, @



      
 ,   
  
*=9+, 3J
F9D1 > ; F
, @ <
%
E 



  
, %# " 

 
= *=9D+, 8J8G
0F81  A F
 , ? 
  


       
 , $
#
&
 
#     
): = *=8+, DDJG3
0F
;3 1 . F
 %,  ;,  A , 7 
     
 

   
 , 
$
# DG * 33 +, 9=J=
0F
6=1 A F

  ;  6 
, @ 
 '   
 




 , $
# 

  8*+ *==+, D8J 8
0;91 K ;
, 



   '  

  ..( 


 , %#
#
  5 # (6 3 *=9+, =GJ 
0;981 K ;
, 
 



 N # 
 /   #
,  

  5 (6  *=98+, DJG
0;<
31 K ;
  <
 ,   " 


  : 

, :
 7 
  
, 33
0;8 1 ; ;U",      



 
 %# #
  5 #
(6  *=8 +, 3=J3
09 1  !, 
  # 
  
 '    
   
%  
 
, 3
# %# $
# 4  5/6  *=9 +,
DJ8
0831  , 


   
, # %# $
#

*=83+, DDJG3

08G1  , @



  
  
 
   
 , %# #

  5 # (6  *=8G+, DGJGD
0991  , 


   
 (

 
 
,  # & #
G *=99+, 88J98
0=G1  
, 4 4 
: 

   , A  , 7 
   -
 , ?
#
, ==G
0
881  A 
F, : 




    
  %# " 

  
*=88+, 9J 9
0981 I   % , 


   
    
   
 , $
# &

 *=98+, =8DJ=93

Section 2.3

The Reconstruction Problem

97

0= 1 I   % ,  
 



    
  
, 
# #
$
# = *== +, 9J8
0=D1 I   % , 



   
 
%
, %# # # ;
#    3 *==D+, GJ 8 
0=91 I   % ,  
 



  
      
 ,

# # $
# D *==9+, =JG 
0=1   ,  



   
  
, <
' 
# %# " 


  * + *==+, 8J3
0=D1   ,  



   
   

 
 , %# " 


  =*+ *==D+, 8DJ9
0M8G1 I M
, 
  




  
  
 ,  
# $
# )9#
  8 *=8G+, 83=J8=
0M881 I M
,  



      
 

  #  
 
 ¾ 
6
, %# #
  5 # (6 *=88+, 9J 9
0991 > K 
,   9   : 

   , A  , 7 

   >
, =99
09=1 > K 
,  



   
   

 
 ,  

  9 *=9=+,  J 8
0=31 > K 
,  



   
  
 #  ! 




  
, %# " 

   *==3+, =JGG
0-6981 > K 
,  ? -   A  6$,  
 
  





 , %# " 

  *+ *=98+, 9J3 
0?891 : < K  ?>  , 



 
, : 
9  ; >
 %   K >  *-+,   
   " 

 , 


,
;, =89
0=31 ; 
,  



      
 , %# " 

  
*==3+, 8J8=
0 <
=91  K 
 $   A <
, 


     ? , %# #

  5 # 6 9* + *==9+, G=J98
0 <
==1  K 
 $   A <
, 


    
, 
  
3  ! 
 %# #  *===+, 8 
0 =81 < 

, 4 



     



  

" 

  &
  &#*#  *==8+, J =
0 3 1 A  
, @ 



 
  6

  , %# #

  5 # 6 == * 33 +,  J=

98

Chapter 2 GRAPH REPRESENTATION

0<G1 6 <
, "   "# 
     
   F
  
 

%
 
 , # $
#      *=G+, =
0<
9D1 K <
M , 

 
 
0<
8=1  K <
#%, < 





 
,   " 

 , 
?# R
% 
  <

  9 *=8=+, 9J9=
0<
1 . <

,  



   
  

, .      


, DG * 33 + DJDG
0<881  F <
%
,     



 
&

 

,
%# " 

   *=88+, =J D
09=1  
, 


  
 / 
 
 
'  
 ,

$
# 8= *=9=L=3+, 38J 
0 8=1 >   ,   % E 
 N    



   K  
 7 <  
 *-+, " 

   : 
 , 


,
=8=
07G31 <  7,   
  $

   , >  *.


+, ?#
R
%, =G3
0I ==1  I , 



       

 
, %# #
 # 
 #  *===+, GDJ88
0> 1 6 > ,  
 '    
, (#  # $
# # 9
*= +, D8 JD8=
0R 9 1 6 R ,   


   




   , %# #
 
5 # (6  *=9 +, DJ DG
0T=91 R T, @  



   
    

 .., %#
. $
# # D8 *==9+, G9J 8
0T=91 R T, @  



   
    

 ..., %#
#
  5 # (6 8 *==9+, 3 J3
0T991 R R T , 



 
&

  
   

 
 

 

, %# " 

   *=99+, 8J 

Section 2.4

2.4

99

Recursively Constructed Graphs

RECURSIVELY CONSTRUCTED GRAPHS

 ! ! 
  

  
     "  
  " 
     "  
 < 

" 4    
  :
 -/ 
  :


" 
 
  




Introduction


  


 


 
  
 
  A!   ,   
    

    &
    
 
 
  

     
 

   
     
  
   ,    #    
 ** &+

*  & 

DEFINITIONS



(  
 

    !    *   ! + 

   
  ,       

  
  
  


  
 
 
  -
  
    
    
!




 
  
  
  #  #  
!


 
 
  
 



( -
 
   


 
  


  
# #     
  
 




 

REMARK

%(


   
    



,   

 
     




  
  
 



     

    
 ,    



  
     
  
       
 


 



    
    &
  <
  3

2.4.1 Some Parameterized Families of Graph Classes
Trees
DEFINITION



(  
  #     
'  *  +    # 
  * 
 
 + ; * +   
 # 
   *½ ½+ ! *¾ ¾+   


  %    &     ½  ¾      *½ ¾+ 

   # 
   ) ½

100

Chapter 2 GRAPH REPRESENTATION

  
 

,  
 * +  A!    

  
6#
,   
!
       

 ½   * 
 + 


    
  


 


 

EXAMPLE

(

4 
   
 


 


   


Figure 2.4.1

%(#- !#(! !


Series-Parallel Graphs

4
  


  
 
 ,  
   
 
      
  


 . 0A GD1  
      4 
    
 
2
 $   
    !      # ,  
,
   
   
 #
  
 


Figure 2.4.2

2!&##&

"""
 ##&

""" 
#

4# , #   


  !     

DEFINITION



(    #             *  +
  !


   #(
"  
 
        *
 )    )  

  +   
 
 
  *  + # 

"  

*     + # *   + 
  
 
 
   
    #     
    # 
  
   

"  

*     + # *   + 
  
 
 
  
    #     #    
    # 
  
    

"   

*     + # *   + 
  
 
 
  
    #   2  # 
  
    


  (  &
%%   
 
    
! #
  #

 , 

   , 
 !  



  

EXAMPLE

(

 
  
  !  
 
 
  
 
  4 

  
 
!

       2 
   #  


  

 


  

Section 2.4

101

Recursively Constructed Graphs

Figure 2.4.3

.!!#! !
!# ! ##&

""" 
#

-Trees and Partial -Trees
DEFINITIONS



(  
'
  
 , . ,     
 #  5 

 *  +



 
  
   

     
' &
   

 
   .  
 ,     



(



    


    


  
 .   


    
,  
  .  
  



  





(  
  


 


  *
 


+  
   


  



(  
     
 
  
  


 
/   
 "   '  
 / 
FACTS

$(
$(
$(
$(


 
 
,  
 
 
  

<
 
 
  
 
  

 .  
    

 
    

 
 


 
  , 
, 

 *
  


 
  


+
EXAMPLES

(



 
  #     4 
 ,   
  
  #  

102

Chapter 2 GRAPH REPRESENTATION

Figure 2.4.4

.!#(! !
&



" &

A
   
      4 
     B
 C   

#     

     
  
  
       
   .¿  !  
'   3  
  ,  # *

  +

'   @
   
    
  
   
 
4 
 ,  
  
 


(

 
      4 
 D #  
 
2     
  *

  
  
+   
  
   
   

#
 


   
!      

  
 

 & 
   !
  
-'  

Figure 2.4.5

 ##&

""" 


&

Halin Graphs
DEFINITION



(  &    

    
 
     
 

     ) $  % , #
 $   
 #   
'  
  %  



        $ 
FACTS

$(
$(

6  
  

   
  
  


   6  
   
 
 %    
 , , 
 
   6  
  
  6  
 
EXAMPLES



(  6  
      4 
 G #, #  

  
#  
 

2 

   
      
   

Section 2.4

103

Recursively Constructed Graphs

Figure 2.4.6

 3
" 


(

 
    4 
 8 #   
2 
'    


 
 
   
        
   
  6  
 2 


 
 
    

 
       
! 


Figure 2.4.7

! 3
" 


Bandwidth- Graphs
DEFINITION


(

 
  *  +         
 '   
'  
       
  !   $ *!+  *+   *#  
 
  =+

(



EXAMPLE

(

 #  
   #     4 
 92     

  #  
 

Figure 2.4.8

)
*&
 
*& 
#

104

Chapter 2 GRAPH REPRESENTATION

Treewidth- Graphs

   #
%  
  <
*, 0 <9G1, 0 <9G1, 0 <=1+


  !   
   
   
    
 

    
     

   
# , #
   %

   
E #
%  
   

  ,  ,  
  >
E

&

,  

    
DEFINITIONS

(







"

"
"

(
(

 


  
   ) *  +   
*






       



 )


  *



 $ +, #

$   
 #  
'  
 




* &+  
    #  * &

 
 
  
    
    $ , 










      

     

     
 

     

(





 
 

   









  '  



 
 







      #  % 
 


      
#   

 

REMARK

%(




 , 
 
 , ,   

      !     

' *

  + @  
, # 
 
  

   
, 
, 

 ,  #
   
  * , 
  #   
# +



EXAMPLE

(


   
   
  

   

     #  4 
 = 4
  
 , ,
      
' 
    ( ½ )  ½    )    
 )    
)   
)    
 )      


 
  # '    
    4 
 =, 

 
   
    
*   + 
    

,  
 
 
#  2  
,  
   
 




   
$


   

Figure 2.4.9

   
$

 #
" &!!#!

Section 2.4

105

Recursively Constructed Graphs

Pathwidth- Graphs
DEFINITIONS

(



  


  

    # 
   

   
            / 
  
'  

 

(
(

 , 




½

 ¾   

    
     ' ½

  

     

(





,    

!  
     





  
 



 



   #  % 
 

       
    #   

 

EXAMPLE

(

   
     #  4 
 3  
'
 !
  




 
    # 


   

  



$

Figure 2.4.10



 
&!!#!



Branchwidth- Graphs
DEFINITIONS





 

$  +, #
 $ 
    &
 

(     


  
  ) *
+   
*
 
  #
 
  
'  '
 
  
 

     



(

$



.  
  
  
' 

    






$ 

$



 

,  
*

  


$  + 


( ; *
+   

 
      
  ) *
+ 
   
     
 

  
  
 '   ½  ¾ 
     $

    , #
 * ½ +  * ¾ + 
  
  


$

(
$





 
$


   

 
    *$  +   ' 





$


 

  

106

Chapter 2 GRAPH REPRESENTATION



(    

     

(







 
       

      #  % 
 



      

#   

 

FACTS

$

  

# 3     

    

$

 

( 0 <=1  
 
   

( 0 <=1  
    

#      

  
 
   
' #  
 

 
/   

$

( 0 <=1  
 
 

  

# 

   

  
#   


EXAMPLE


(

 

#  
   #     4 
  * 
  

+2  


        


Figure 2.4.11

)
*& 

 # 
&!!#!

-Terminal Graphs
DEFINITIONS



 ,  



% *# ++   
!    #
, #
 
   

(      ) * $  +   
'     
    
 $ ) ½ ¾        , #
 $  
(    
 

 
  
    ! 
  + ) ½  ¾    

#   # 



 

EXAMPLE

(




  
 
  
   #  4 
   I

 

  



 #
    
 
 2 
  
 
  

 



Section 2.4

107

Recursively Constructed Graphs

%(#- !#(! !
&
" 


Figure 2.4.12
REMARKS

%


(


, 


$

)



 #

  

     



 
      *,

%

(

   


 






½     



%+
*


 

+ # 

% # +
*

+, #


4




½ 

(


    
(


#

 

*

+ )

 ¾
*

+

 
  $ 

$ ½

, 

) *


 $ 


 

 ) *



 +,



+  # 


 



)

 * 

 + 
    
 (

   

* +

$ $  $

 
 
 
(

½  ½  ¾  ¾  ½  ¾   ½


"




½   ¾    ½



 
  



"

 $ 


 
   
' 



"

+  

 
     &  
'  ;

      
"


  
  *

+


 ,
  
   
 
 #





   



  
 
  
   
 
       ,   



)

*

+ )

 ¾
*

+

$ ½ $½  ¾ $¾

 


(

% +*½  ¾  + &  *½ +  *¾ + 

*

 

.



    
'  , 


  

  * 



+  
 #


 ½     
*

+ )







     




Cographs
DEFINITION



(




  !



   #(

"

 
  #     
'  

 

"

.

"

.













 ,    & 




 ,  





  %  








  


 

 



  







 ½  ¾
½  ¾

 

 
 

 , #
 

     *

½ ¾

+ #






:
  
 


  


  
  



108

Chapter 2 GRAPH REPRESENTATION

EXAMPLE

(



 


   
  4 
  
  

  
   !  
    
    
 *+ 
 
   #
 


Figure 2.4.13

.!
 !#(!

FACTS

$
(
$(

0:; 91 
   

    

 
0:; 91 

  
 



Cliquewidth- Graphs

 
  

. + # 

  0:- =1  
   


    % 

  
  
  

DEFINITION



( ; 0%1     
 
!


   #(

    



#
   



 #   *+ )   *+ 01  
 / # % 
 
. ½  ¾ 

 / #  
   
01, 
*+   &    ½  ¾  
 / #  
 
* +  
  *½+  
 / #  
 , #
 *½+  
 
 ½
     *½  ¾+ 
  *½ + )  *¾ + )

*+  
  *½+
 
 / #  
 , #
 *½+
 
 
 ½
 # 
   

 #    


"  
 
"

















REMARK

%(

A!   9 !   
 / #  
  
 / #   

          
    
 / #  
  
 / # 

    
 
   
 
 
  



   , 




    , 
' 
 ,  
     
 
          
  , 
, 

 
  

 


  
 .   
 ,  
 . 

  
   
 
  
  .  

     %,
 /   !   
(

   
#  &
 

   
 

Section 2.4

109

Recursively Constructed Graphs

  
 -
 
   
#  
 ,  
#    
  #


  
  
 
  
 <  
, 

 /   
 / #  
 , 

 / #    
  # 
  
  
 
 
 /     

   

    
 . + * 0:@331+
EXAMPLE

(


 / # 


      4 
    -'   , 
  
  
  !  
    
    
 *+ 


   #  


Figure 2.4.14

 "4(*& !#(!

-NLC Graphs
DEFINITION



( ; 0%1     
         #    
  
 
 0%1  0%1  '( $


 %   !


  
#(

 #   *+ )   *+ 01   %?;: 
 
. ½  ¾ 
 ?;: 
   
01,   &  ½  ¾   ?;:

 , #
 ½  ¾  
 
 ½  ¾      *½  ¾+ #

½ ½ , *½ + ) 2 ¾ ¾ , *¾ + )
 * 
+      
 
  *½ +
  ?;: 
 , #
  
 
 ½  # 
  


 #    


"  
 
"







"





EXAMPLE

(

 
   4 
    ?;: 
  . 4 
 D,  
   

  


     

 , ,
, , , ,   #  
  

# 
   % )    #   
  
  ! #  

    
,    * &+ 
 # 
   * 
+ 
    
!

 
  
  
   

110

Chapter 2 GRAPH REPRESENTATION

Figure 2.4.15

 &25. 
 !#(!

-HB Graphs
A  B  
,C  
 

   

   
  
   

  
    

DEFINITION




( & $

 
   %   
 
  
#
 
  



*¾ +   # 
    
 


   

 / # * 5  + 
 
   
REMARKS

%(

 # 
   

  

   
  
 
 

   
   
,   
    
   
 
  

 
 
   
,   


   

     



%(

  
 / #  
       
  ?;: 
   , '
 
  
'        
 
  
  


%

(  
 
      
' 
    
    
  
  * +

%

( 
/
   
       
  



 ,
#        
 / # 
    


    
$
  
  6 
  4

     
,  0K < 3 1

%
(

6#
, 6 
  
      !
     

  

    
    
 @  
,  
   
  
  

 


 / #  
      
  , 


 / #  
    6 
 

2.4.2 Equivalences and Characterizations
Relationships between Recursive Classes

  
 / 
 
 
   


  
 
 ! 

    
  <
   
   # 7   
!

 

Section 2.4

111

Recursively Constructed Graphs


,   
  


 


 
4
  
  = *

+  0
;< ==1
FACTS

$(
$(
$(

 
   
#            
  

-
 #  
    #      
#  
 


  
  

  !   * 5 +
 


  
 

 * 0> 69G1, 0> 981+

$(
$(
$(


 
 
        
#  

 
 
 
 
  #
   &
%%   
   

<
 
 
   #
   
  
  
  
 



  
  
 
 
 

$(
$ (

<
 
   


  
 
  
   
#  

6  
  

   
  
  
2  
  !
 
  
  
    



 
     
 

$
(
$(
$(
$(
$(
$(
$(
$(
$(

0:- =1 :
  


 
 / #  
 
0:@331 -
 
#  
   
 / # * ½ 5 + 
 
0 <=1 -
 
   

#      
#    ; 
0 <=1 -
 
   
#      

#   

 5 

0>=1 :
  
 '
  ?;: 
 
0>=1 -
 
#  
    * ·½

+?;: 
 

0>=1 -

 / #  
    ?;: 
 

0>=1 -
 ?;: 
   
 / # 
0K < 3 1 -

 /

 
 
#  
    6 
 

Characterizations

<






"  


  
 
 
 
   
 

   
   

DEFINITIONS



(  
 
 , 




 
    
   ) *  +
  
 

, #   *
+ ) 



 *
+ ) 




(   
    



  
  #     

  #   
'    
 
 *B
 C+  

 ! 
 ,
 
   #   
' #
  

    &

  


 
 

, #   

   
   



(  
  0   
   
       
    
  
 !  / 
  '

   


   
 

112

Chapter 2 GRAPH REPRESENTATION

REMARKS

%( 
   
 !

&

 *    +  F >


 # ( <     
 
 #  
 
        0

     
 ,  0   , 

  
 
 

 
 '   !   0½ 0    0  
 ,  
  
 

         
    


  
0 
   
%( 
  <
*0 <991+
!
 >
E
&

  # 


,     
 
 
 
 

 

 "
     7
 ,    
,    ,   '   2
#   %#   
 
   
 

 "   
 



%( 
  
  
 
 
 
   ,   

<

  
 



"   !    
   


%(  
   
 

  
 
 %# * 4
  #+,  

     ' 
  
 

  
 
  %# 
  

  

FACTS

$ ( 0:; 91 :
    
  1 

$
( 
 
 
     .  


$(    
   
  
  
    
    

  
 ,

. 

$(  
   
   


  
 .  .
 
$( 
  
  
  

   
(
#  4 
 G

.   
 
 

Figure 2.4.16 $! !# ! 

" &#

2.4.3 Recognition
. 

  
 
 


 
  


,   

,   

  
  
  #
 
 " 

Section 2.4

113

Recursively Constructed Graphs

REMARKS

%( <

  
 
 

   

   4
' , 6 

 
 

 "  !
   


        

  


    * 
   , 
   
+, 



   !   
     ,
     
 
 
     
    
 * 0:? 91+

%( 
  
 

 
 
  


 ",    ,  



   
    #  



 * 0A GD1+(
 
 
 
'  
     
  *   +  *   +   # 
*   +2
 
   
         2      
  
   
'  
     
     

 ,       
   

   
 ,   
  
  
    
  
2 
# , 

 #   #   
.

 
  #   .  


%( <  


   
     
  
  
 

 * 0

9G1+  #  

  
 *0<=G1+

EXAMPLE

(   
    

 

  / 
  #  4 
 8

Figure 2.4.17 %(! !
!# !


" &
Recognition of Recursive Classes
FACTS

$( 

 

 "  

   


   
 
$( <
 
 
 
 

 "  

   


 
 
 
$( 
# , # , 

# ,  #  
 
 


 "  

   


 


  ( 4
!'







*

 

 

+  

    
  4
 G 


$(  
 
  4
 G
 

 "  %
 
!' 

  ( 


   



   4
 8 
 



 
  

     
   


$( >   , 



  
 

   
    

 
 
  4
 G * 0=1 

$ ( >



) 2 0<=G1 #



) +

  
  
 
, 

  
 

#   
   4
 G 


 



$
( 

# 
  
       


  0<=1

114

Chapter 2 GRAPH REPRESENTATION

$( < 
 
  
 




"   !    
   
, 

  

 " *
0 <991+


  ( 4
  #     
   

   

  <
 6#
, 
   '  




    

  '         

    

$( 0> 981 -
 
  
    
# ¼ 
  
 ¼   
    


 


 
     
  4
' ,  
  '       
 ,  ¼  

  
  
 B
C

    
  
 4
' , 
 
  
   
 
$( 0:<9D1 :
 
 

 "  

   


 
 
 

$( 
 '    

 " 
 / #  
    
$( 
 '    

 "  ?;: 
    
$( 0K < 3 1 . 
  6 
 , 
 

 ! 
 

  # 
 
 
 / #  
   , 
  
  

  



 #


 
   
    # 

  

    
    
 / #  
   *+  
   

 
6 
    
  

 

 / #  
   #
  


References
0
:
981 < 

, A  :
 ,   
%
#% , : '   ! 
    %
,  $ %#     
$
  *=98+, 88J 9
0
:
<=1 < 

,  :
,  
%
#% ,  A <,  



  
 

 , %# $ 
*==+, JG
0
6
=1 < 

, < 6  ,   
%
#% , *+, -Æ
 

 
  %
,  
    
  $
#  *==+
0

9D1 < 

   
%
#% , :


"  

    
 
%
,   & 
  *=9D+, G=J8D
0

9G1 < 

   
%
#% , :


"  

    
 

,  $ %#     
$
  *=9G+ 3DJ
0

:=31 < 

,  
%
#% ,  A  :
 , 4
   





"   
  
, 
$
# 
*==3+, J=
0@=91 ;   < @
, @  


  
  #  # 1 , 

  $
#  *==9+, J
0 81 ; >  %   -  
, 
 
  



"   %
,
$

  *=8+, JD

Section 2.4

Recursively Constructed Graphs

115

0 <=1 A  
%, ? 
,  A <
,   , Z
%
'
    
, %# #
     (  *==+, 8J 9
0991   
, 

 


 
    ( 
   


, A A 
 , <
  .
   : 
<

, 
 
.    
, =99
0=31 6 ; 
, :  
  #    
# , 
# : # :))
=2//, A 
  : 
<

, 7

 7 
 ,  ?
,
==3
0=1 6 ; 
,  
   
  
# , 
  
 
*==+, J 
0=G1 6 ; 
,   
  
 
!  
 
    
 
# ,  $ %# 
  *==G+, 3DJ8
0 6F=D1 6 ; 
, K   
, 6 6 ,   F%, 
'
  
# , # ,          
  , %#  

 *==D+, 9J DD
0K < 3 1   
, K ; K, I ,  K < 
,   
    
 
 / # % 
 ,  
 * 33 +
0F=G1 6 ; 
  F%, -Æ
 


  
 

#   
#   
 , %#  
  *==G+, D9J3 
0M=1 6 ; 
  6 M

,  #   
#  


 ,  $ %# 
$
#  *==+, 9J9G
0=1   
,   
%
,  , :, A
  

   



  
 
,  $ %# 
$
#  *==+, 9JD3
0
;< ==1  
M
, I  ;,  K < 
, " 
    9 , <.

   A 
    
 , <.,     *===+
0:=31  :
,  




 
  .( 
 "  
!  
 ,  #  
#  *==3+,  J8D
0:= 1  :
,  




 
  ...( 

   ,
 
, 
 '   ,  #
>   #  *== +, D8J 9G
0:=D1  :
,  




 
  I...( @
 , 
      .  *==D+, 3J
0:=G1  :
,  




 
  S( ; 


,


 
  
*==G+, 98J
0:- =1  :
, K -
,   "
, 6
#
   

 


, %#  
 
     *==+, 9J 83
0:F 91 A  :
   A  F
% 

%, 4   


  ! 



 ,   & 
  *=9+, 8J G

116

Chapter 2 GRAPH REPRESENTATION

0:; 91 A  :
 , 6 ;
,  ; < 
 , : 



 , 
  $
#  *=9+, GJ8
0:=1  :
   , 



    


  
 ,

 
  
*==+, =J9 
0:? 91  :
 &, A ?,  >   %, 6  
   

  
, $
    *=9+, 98J =
0:@331  :
  < @
, 7 
   
 / #   
 , 

  $
# 
 * 333+, 88J
0:<91 A  :
 , R 
,  ; F <#
, :
 (

  ,  

   
,   & 
  *=9+, =J D9
0:<9D1 A  :
 , R 
,  ; <#
,   


   
 


 ,  $ %# 
  *=9D+, = GJ=
0A GD1  K A Æ,    
 
 
 , %# $
# # # 
*=GD+,
3J9
0 ==1  :  
 7 
, @ 
 / #   

 
 
,
. 
 &
  
  GGD *===+, DJ8
0
<%=1 A 
  A <%
F , ?: 
 


 "  
  

 
,  $ %#     
$
  *==+,  JD
06R981 S 6  R R, 


    
     #
 

 
 
 ,  
  
  *=98+, DJ9
0F.79D1 R F&  ,  . " %,  <  7, :


"    
 
%
  
 

  


,  8=J9   #  ?=-, =9D
0F=1  F%, 
# 
    
'  , . 
 &
  

  , <

I
, ==
0FF
=D1  F%  A F

, 
#  

  
  
 , %#  

 *==D+, GGJ 9
0=1 K  [%   , 
 
!  

    

 , %#  
  *==+, J 
0
=1  
%
#% , 
 

   
 /  
!     
 
  
,  D=JG33   # $ @ 
  " 
$ , < ==,
: 

  < , 
  <


 
, ==
0 = 1  , 4   
'   

 
   
#  /
%, 
J 9   # A
 

  
  
  , == 
0 =1  , 
#   (  ?# :
   
  <   

 ,  98JG   9   
  +AAB,     , :

7 
  
, ==8

Section 2.4

Recursively Constructed Graphs

117

0 81 A K , @   



"   %
, 
$
#  *=8+,
8J 
0 <91 ? 
   A <
, 
   
 . -'
    
, %# 
#
     (  *=9+, =JG
0 <91 ? 
   A <
, 
   
 ... 

# ,
%# #
     (  *=9+, =JG
0 <9G1 ? 
   A <
, 
   
 .. 

 
 

# , %#  
  *=9G+, 3=J 
0 <9G1 ? 
   A <
, 
   
 I -'
    

 , %# #
     (  *=9G+, = J
0 <9G
1 ? 
   A <
, 
   
 I. A &   

  
, %# #
     (  *=9G+, DJ9
0 <991 ? 
   A <
, 
   
 I.. A &    


, %# #
     (  *=99+,  J D
0 <991 ? 
   A <
, 
   
 SS >
E
&

,
 
, =99
0 <=3J1 ? 
   A <
, 
   
 .I 
#   #
/  

, %# #
     (  *==3+, 8J D
0 <=31 ? 
   A <
, 
   
 .S A & 
 ,
%# #
     (  *==3+, 3J88
0 <=3
1 ? 
   A <
, 
   
 I...  F
#% 
 

 

, %# #
     (  *==3+, DDJ 99
0 <=1 ? 
   A <
, 
   
 S @

   


   , %# #
     (  *==+, DJ=3
0 <=1 ? 
   A <
, 
   
 SI. -'
    


 ,  
, ==
0 <= 1 ? 
   A <
, 
   
 SS.. .

 

 
 %
,  
, == 
0 <=1 ? 
   A <
, 
   
 S. A 
   

,
%# #
     ( 
*==+, 8 J3G
0 <=D1 ? 
   A <
, 
   
 S...   &  

, %# #
     (  *==D+, GDJ3
0 <=1 ? 
,  A <
,   , Z
% '
    

 , %# #
     (  *==+,  J9
0<=G1 A  <
, @  


    
#    
,  $ %# 

$
# *==G+, 3J8

118

Chapter 2 GRAPH REPRESENTATION

0< =31  <
  ;  , 



"   
  
, &
' 

*==3+, ==J 
0<
991  <
\
, > 
     
# V,   # 1
 
  
$
#  
, 
%  F/ *=99+
0<=1  A <
  , :
    



, 
 
 *==+, 8J 
0>=1 - >%, %?;: 
     
, 
  $
#
 *==+, DJ GG
0> 981  I > 
, ; 

  %
  
 , A A 
 , A

  : 
<

, : 7 
 , =98

119

Chapter 2 Glossary

GLOSSARY FOR CHAPTER 2

 ) *  +(  

  
 ,  

 
'   2 

 
' , 
    
    %  

    


&
  


 "#

  J 
 
  
 
 


 


  J      
  
 
   ) *  +( 








' , #
 0 
1 )   
    
 
'  
'
,  0 
1 ) 3

# 

-#
 !#(! ( J   
 
 


 




  
  
% 

(       
 
 
    / 




""&
# #!#&
# !"( 
    
  # 
 
 

   
 

"" !#(! ( J   
  (   



   


1  J 
    
   

 
 (  
   &   
'
 
 





*& 
(  
  
#
 
 '   
'  




 




 

! 







$

*!+ *+






(



 
(  
  # 

   #   


&!!#! J   
   ) *  +(  
*$  +, #
 $   
 
#
 
  
'  '
 
  
      &
  
 
  $   

, 

"(  


     #
  
  
  
' 
$    


*& 
(  
  # 

#    

 

* J   
  (      #  % 
 


   





&6# #
(  
 
!   

   
  



 
    
 
',       
 
'  
        

   
 



' 




&6# ( 
   



 
    
 
'   
 

  
!
 



!
" (
"! 

 (  


     

  / 

*+ ) ½  ¾         

     
 , 

 # 
   ) *½       +  0 ) *!½    !   + 
 

$
  &
    ! (
)        
  
 


!
" ( J   
 (   
   
'

  


  
 


.7(      
#
# 

6
 ! #!!#(  
  


½ ¾



, /*½+ ) /*¾+ $

½ ¾

/


  
 # 
 




!
" 
(  
  
   


   

 
"
## &!#(! ( J   
    
  (    

 
    
%   #
, 
#   
  

  ,  
     /  
 

  

120

Chapter 2 GRAPH REPRESENTATION

  
 (    

 
    
%   #
, 
#   
      

 ,  
     /  
 

"
## !#(! ( J   
 

"
*&  
(  
     
  
  



.½ ¿



"4(* J   
 (       
   
  Æ
 




 
  
  

, #       ,   & , 
 
 
 

 
( !


   # *0%1     

    +(
 
   #   *+ )   *+ 01  
 / # % 
 
. ½  ¾ 

 / #  
   
01, 
*+   &    ½  ¾  
 / #  
 
* +  
  *½+  
 / #  
 , #
 *½+  
 
 ½
     *½  ¾+ 
  *½ + )  *¾ + )

*+  
  *½+
 
 / #  
 , #
 *½+
 
 
 ½
 # 
   

 #    


"4(*&



"
"

















!
( !


  

"  
  #     
'  

 

½  ¾ 


 ,    &    ½  ¾  

 
. ½  ¾ 


 ,  


 ½  ¾  

 , #
 

  %      ½  ¾      *½ ¾ + #
 ½ 
 ½  ¾   ¾

" .
"

!"!
(!!# !" .7( 
  !     

 

  
 



  
  , #      
   


     

 

!"! "
## J 
 
 (     

  
    



(    3 (  % 
  
'     % *
   
+2 
 ,  
   3 ) 0%½    %1   
'  
   

!"! J   
 

, -
" J 
 
 ( 

     

 
 
'
!"!&#- 
(  
    
   #  %

 



    
     %

 

  
 

   J 
 / 
  ) *½ ¾    +  
  *  #

 $
 
   

+(  / 
 ) *½ ¾     +   
  
 * 

   
+ 
  * +     )       * +  )  2
  


        *  +

!- !



!##  J 
    
   

 
 (  
   &  #



  
  


 

  
 


1 J   
 



( 

  *+   
' 




 !
- (   
 

 &
  
 #4( J   
 

 
  



(  /


    
 


  
   

  , #


121

Chapter 2 Glossary

 -!

3

J   
 





5

6



# 
   
&6# !#
&6# #

) *

* + ) 0

) 0

%½    %

1(  

 

4  %½     4  %
* +

* +

1

+(   #
  

    




 



¾

(    !
 
 
    !
 
   
 
(  
 
     

   
    

    
',
%  

   &
 
', 


 
  

 
 
  
'

&6# 

(


 
  
   

    !
 
  


 

&!
!

J   






  #  
    



  
 



 

) *

+(   
  
   

     #   
'   
 
 *B
 C+



,  
   #   
' #
  



   &

   

 
 

, #   

   
   

&1

&" #(

 

J   
 

( 

   *



J   
 





+     
  

(  
 




  


2 
 %   #

&&" #(

  

&
!

J   
 




J   
 

   
 



&!8
" "
##

) *

 



(   
    






( 
   
  
 , 
 
 





  



   

&!#(" 






(  
 





  

  

+(   
  
  









 

, 


# 
 



   




&!#(" 


#  

 





(  
  

 
 , 
 
 

, 
 



  



   












&%!#(! .!(

( 
&

  
 
     

  



 

&!#(! (

% 



J   
 

#
  
 







(    
  
   

 /  
 

&!#(! !

 
0
&!#(! !" !
#((   
(  
 

 *


#    
% 
*


  ,

 
    
 X(




 
   

&&!#(! !"

 / ,  

X

"
 

-
-&1





+ J #
    

 /    #




 

 
    
 X

( 
  
  

 ,  
  
 


 
 



 / ,

 


  
  
 




(  
   #
 
    
  




J   
 



(  
' # 
  

J   
 







( 

   
 

 



 







122

Chapter 2 GRAPH REPRESENTATION

-&!#(" 
(  
   

 /  
    


' 
%

$"!&+
#
""
"!(  
 
   
   *


+ # 
'

 
'
, 
 

 


!*
  J 
    
   

 
 (  
   &  


'  
 




 #!!# !" 9':( 
 


   Æ
 

   #
#   
  
 



&!( J 

 (  




# 

  #
 
 

3
" 
( 

  #  
  
        
,

#   

  
 ,  

 
       


&3)


( 
      
  

    # 

 

    
    2  A!   3   

½     , 
  
     

    
%

""
 1( 

   
 

 
   
 



 

 


""
 1 !"( 
  
  #

   



   
     
%   
 

 ) * +     
'
, #
 0
1 )  
 
    3 
# 
 


  J       
   ) *  +   


' , #

   
 

 
' 
0
1 )   
 



' 

 


  J      
 











9':( 

3

# 
#


# 
# 


  0 , 
  
   
   0
#!!# ! "
" 
#   0 (  
  , ( 
0 , 
 


   ,  

   ,*+    
#!! 
#( # 
 







#!!# ! #" 
#(  
'  &
  

 &


 

 

#!!#&!" !"( 
     /  
/" "!#( !
# ! ##

(   



)

½








'$

/"0#
"!(  
 



  
 
'
   


  # 
 
 

    
 

"
" 
(  
  # 

 L
 
 ,   # 
  

 ,    
  !   

"
&
"!( 
 
 

 +

*

! J   
 
%( 

(



* 5 +   


 
  
    

 
 

 
 

   !  /

)


 

%  

!("( # 
 
    
 ,    

  
 '
  

 
       
 

!!!# * !

   

(   &
   



0

J     
   , #
    

   
    
* +  

! 



123

Chapter 2 Glossary

 !  

0

! 

 !  

 * + * +      ,    
    
  * + * + 
       
 
  
 
#  
 
   0 1 

0
0

!!!# J     
 



 0

0
 !  
0

 



 (    
  ( 

   
     ,   * + * +        
 
  
 
   0 1

((    



 


 
  
  
   
*    /  
 
B  
 , C+

! 

 0

!!! J   
'

 

4 *+



   
 (

0



    

 &
   . 

&25. !&"
"&!!"" 
( !


   # *01 
   
      ,  #    
  
   01  01+(
"  
   #   *+ )   *+ 01   ?;: 
 
" . ½   
 ?;: 
   
01,   &       ?;:

 , #
     
 
         *  + #

  , * + ) 2  , *+ )
 * 
+      
"  
  * +
  ?;: 
 , #
  
 
   # 
  


 #    

!# 6
(!
!(  

 
  *   #     +

#   
 

,        
 ,    !
,     

2&!#(" 
(   
 



 


 $

 


     
 *   
 *  +
  *  ++, 
 

   ,   Æ
 
   
 
 / 
! J    
  


    *
+   
 
) *
+(
  
 

 
  
 '     
   
 $

    , #
 * +  *  + 
  
  









$
$ 
 

$




" &(  
    


 

(  / 
   *  + *  +    *


&!!#!(  

    # 
   



 



+




*& 
(  
    #   

 

* J   
 

2 



(   #  % 
 
    
 #
  
 
  

 (  
   #
 
 
  
  


 
/   

 "   '  
 / 


" - !
(  
'   '    
 
 



 


!"!
" 1( 

  *  +  




   

 
    
%

!"!
"&
"!(  
 
 

** 5 + +   




 ) * + 

 
!  J   
  ( 
 
  #
  # 
   
 
 

  ,  
        # 
  


!8
" "
## ! 
#( 
   
  
 , 
  , 




       % 

 

124

Chapter 2 GRAPH REPRESENTATION


!#(" 

(  
  


 , 
 
   # 
    
 

, 




      

 


!#(" 
( 
 

 # 





   





%!#(! .!(( 
&

  
 
  #    



 



 

!#(!  J   
 X(    
 
  
 
# 





,  



 * X  +  



 



(    
  
    
%
   /  
 

 (  
  0 #    
%  

!#(! ( J   
 


 #
 


 

!#(! !

(#-" !#( 
 "
##( !    *   ! + 
  

 #,       

  
  
  


 

 
 
 2 
  
    
    
!


 


  
  
  #  #  
!


 
 
  
 

6 !

 !"! J   
      #

  


 ( # 

 #    

   #

   
     
   
  
 



("
 ##!(    

  
 
      


 ,

 ,  F



##&

""" 
 #      
   ,  *  + J !


 (

"

"

"

"

 
 
        *½  ¾ +   
 
 
  *  + # 
 ) ½   ) ¾ 
 

*½ ½  ½+ # *¾  ¾ ¾+ 
  
 
 
   
   ½ #  ¾   
    # 
  
 ½  ¾

 

*½ ½  ½+ # *¾  ¾ ¾+ 
  
 
 
  
   ½ #  ¾  ½ #  ¾  
    # 
  
 ½  ½

  

*½  ½ ½+ # *¾  ¾ ¾+ 
  
 
 
  
   ½ #  ¾ 2  # 
  
 ½  ½

# &!#(" J 
  
 

(  
  
 



 
  
     

 
    
%

# !#(" J 
  
 

(  
  
 


 

  
     

 
    
%

#
# 
  ) *  +(   #
  

        

#
"8
! !
!"! 3( 

 
  
 
  
!


    

    .   3 

#
" !"!(  
 

   
  
!  
 

#(((  
 * X  + #
    !  , X   
  
   
 


,     

 



&
" (#- 
( 
        
 


 
 
,  
      
        
   

   
   
  *< A!   D   +

125

Chapter 2 Glossary


#- "!#( !






       

 

(
,



(  
 






    *









+    





 
  #  

,    #      

(#-" 6(

 
  #     
'


  &   
  # 


&


*


  !+(


 




  
 

.





 


2 
,  
 

2 
,  
 


 
 



     
' &
   

    

.

 
 ,     





"(   
    

&!!#! J   
  
,


"










        

 )



" 
  *
" 
 


* J



* & 

 


+



  
 

) *



 
$

+(


   

, 

 
*



$







+, 
 

  
 #  
' 




    


* &
 $
# 








, 






 

 



(      #  % 
 

    

2 
 #
  
 
  


*& 
(  
  # 
#    

 
-&" #(
 J   
  (  
       
    
 
'



    
  2 
 (%   #

&-&" #(



J   
 



(   
    


  

     
  

*
1" &!#(" 
 J
   
 (

 



  



*½
*¾





  

 
 

#
    
   



*
1" !#(" 
 J
   
 (



  

 
 



 
 



#
    
   



J   

 
    *
J   

    *

$ 
$



 
  






+(  '   

   







+( ' 



  

$



Chapter

3

DIRECTED GRAPHS
3.1

BASIC DIGRAPH MODELS AND PROPERTIES
   
 


3.2

DIRECTED ACYCLIC GRAPHS
     


3.3

TOURNAMENTS
  
 
 
  

  
GLOSSARY

Section 3.1

3.1

127

Basic Digraph Models and Properties

BASIC DIGRAPH MODELS AND PROPERTIES
   
 

 

 
 

  

   
  

 
 


Introduction
     
 

      
 !

  "


 
  
 
 "#
     
 
   
  

    
 


 
 
    
   
    

 $ %&'(!
$)*&&(!
 $+,(   "
 -     
  $
),(

3.1.1 Terminology and Basic Facts

  
            
     -

  
 


 
.! #   
   
     
" 
-
!

 
     "        
 /  0  
 1
   


 
     "

 

   
  
" 



Reachability and Connectivity
DEFINITIONS

2

3
 
!
  
  ¼   


 4

 5 ¼  ½ ½  ¾  

½ 





 "

!  
  / 0 5  ½

 / 0 5      5   


  #
6  
"

"  
 





  


2
2

 
     
   
-   #
6 4

    
 
 
 #  
 
    7
#!   
           


2

   " 
 
  
 
 
    
    
  
 - #
6

 - #
6 7"
"  

 


   
 /  "
#
60
%

 

128

2
2

Chapter 3

DIRECTED GRAPHS

  
   
     " # "
 






  
  
 
  


 
   
   74"

!
     
 
  



  




 "



2 %                 
 
      
    
  
  # "   5          !  
  

   
    "   "  
 
  

   

  
"      
"      
EXAMPLE



2     #
 
 !        !      !
 
 
   8  
  "-          
 
 "-  
 
   -           



        
  
   
   
  
 ! 
#   -        
   

Figure 3.1.1



  

  
 
 

FACT

2

%  
 
   
-


 
  
 4"


    !
         
 
   
 
  4"
 
   
 
Measures of Digraph Connectivity

+  
# 
 
     
 
 "
 

  
   "
 9
 9:    


 

  
 
     ; #  # 6 /0
  



   

   
 /'90
DEFINITIONS



2    
 

  
  
 # 
 
  
"!   /  
0
 

Section 3.1

129

Basic Digraph Models and Properties



2     
 
   
  5 /  0 
" 
    
  "-
   
        
 !

 
  /  0 

      
 
-
   

       
 



2         
 -"  - 
  
  5 /  0!
   /0!    < 
"    
     
 
     "
 
 /  " 
 
 -"
 
   0



2  
     
 -"
 
!    /0   
< 
      
        
 

 
+     

!  "-
  - "
   
 ! "
       
 
  
 ! "


   =
 
> 


  
      "-
 
   
         -
 
   
 /     
 0
 
 
Directed Trees
DEFINITIONS

2
2

    
 
 #  
 
 


    
  
"
  " ! 

 
 !  
   "  " !  
 - 


  
.
 

          
 

 
!
  

!

REMARKS



2   
 
 
   
 
!   - 
 
4

2

 

  
      


6 
  

Tree-Growing in a Digraph


 !  # 
#!  
 
 -  #
   /
 0! 
    
 3 !
  
 ! 
  
#  "
 


    
 "  
  
  
"
  /! -#
0!  "         





   
 
DEFINITION



2      
    
 
 

 #  

  

#  
      

130

Chapter 3

Algorithm 3.1.1:

DIRECTED GRAPHS

!  "
#$
%   


2
 
 


 "    
!2
    #   



 "-

   
3
<  
 " 
+

,  " 
3
<

   25 
+
  


   
    
 
 
      
%  
 /0 /#
    0

 
 "     
+

  " 
 25  ? 
  
 "-

   


    
+
 
     
  

  

  "   ! #  " 6 #"     
   
   #   

EXAMPLE

2

    #
 



 
   
  
 
  

   @ 
 "
 
  
 

    #  
    
  
   + 
 

 " !    
   
     # 

 

 "
/


   ,0 8  

 # 
     #  
 "- 
"   

 






Figure 3.1.2



 
&&  '&  


FACTS



2 %
   # " 
 
  
 
   
  
     
 
    
 
   
 
 
 "

 "  
"  
 "-



2 3   
    
 !     

  
  ! 

   
 "
REMARK

2

7

 
 " 



  
   # 


Section 3.1

Basic Digraph Models and Properties

131

 
2 #
 -  # 
  
  "


         
! 
 
      
 .
@ #   

  7

     -  # ! A


"   /,0!  A       
 
   


  
 
    
        -  A
! !  ! $
)&&(! $)*&&! 9(!
 $& (
Oriented Graphs
DEFINITIONS



2   
 
 
 
   
 
    

  
  
 
 !
  
     
"    

 #
 
  "! # 
 

   
 


 



2    
   
 
 
 !  
  
-
!

# " 
  "!   


      "  "

  




2  
    
     

    
   -    
  
  
   
 
EXAMPLE

2

.   
  #     ! 
 
 ¾   




Figure 3.1.3

(&) 
 ¾  
 &) 
'&

8  
 ¾   

   

 
     
"
-  3

! 
  -  


 Æ     

  
 


  

<
    


 
 #
  " 
B7    & &
FACT



'' * "
 $ &(    
    


 


2

  
  - 

Adjacency Matrix of a Digraph
DEFINITION

2

       
  5 /  0!    !  " 

 $   ( 5

  
  
   5 
   
-

   5 

132

Chapter 3

DIRECTED GRAPHS

FACTS



2   #- 

1
 
 4
         "!


- 4
   



2 %    
 #
1
 
     "
     $  (
    #  
  4
     - #
6 
  
EXAMPLE

2


1
 
    
    9   " 

  
 
 


   
 '! " 
     - #
6
4
 ! #   / 0-   



5

,
 
,
 


Figure 3.1.4

,

 




,

,
,

,


,
,
,






 
 
+) 
,

REMARK



2   
 
  
 
    

 
#!
#  



  
!
   #


   "
/ 0
  #   
 

   

 

    

 
  '9

3.1.2 A Sampler of Digraph Models
3   ! # 


#    
  
  
  
  

      
Markov Chains and Markov Digraphs

    
6 "    
 
  


6 #
   
  
  !

  


 
  
 # 
-
 


 
  


   
 
 

   

   
1  
 

      / ! $ :C(! $+&9(0
DEFINITIONS

2

 4  
  "


 !  5 ,     ! 
    
 5            


  5 ,     !
   

      
  "

   3 

!
 
   

-


/ 5   5  



5        5  0 5 / 5   5 0

Section 3.1

133

Basic Digraph Models and Properties

2

      
A 
 
   
  





 !       / 5   5 0 5 
   

 







2  
 
  5 /  0 

 
  # 
-
 


   

  
 
 # "-  5  !
-  5   ,!
  

    
    

  

2

      

6 " 
   
 # 

   

  





  

EXAMPLE

2

 

 

2 

 
 # D
 
  

#

#  
   3      
!   # D E #! 

D B 
 
 


     

 



 DC
%   
      

  
! #  5   
 

  5 ,    9 C!
  4   
- 
6 " 
 

  

 
6 "  
    
6 " 

  #    C

,



 ,
 :C
 ,
 ,
,


9

C

,


,
,
:C
,
,
,

Figure 3.1.5


,
,
,
:C
,
,

,
,
,
,
:C
,

9
C
,
,
C
,
, C
, C
,
C
,








$'&
* 
   
, 
-
./



Equipment-Replacement Policy

+ 
 
  

 
    

 
 

<      #
 


  
-A   

EXAMPLE

2

  
  
F   
# 
  D'!,,,!
 
   #



 DC,,   
      
   1


  
 

"
   6  

  #   

 
#  
   !


  

   
      A" 


.




D',, /   
  
0
D&,, /   
  
0
D,, /   
  
0
D',, /  9 
  
0
D,, /  C 
  
0


 G

D !,,, / 
- 
-
 
0
D!,,, / 
- 
-
 
0
D&!,,, / 
- 
-
 
0
DH!,,, / 
9- 
-
 
0
D'!,,, / 
C- 
-
 
0


 -
&2   
 
  "!

     '!  

$


 
     '     
 '  A     


134

Chapter 3

DIRECTED GRAPHS

    
 
  #  !  !

  
#   "   " 


 
#  " ! # "    
    

# 

 
   
 
 6  
    
   !
" 5   # 

    
 

?   
     
   ?       
 
 "

    
 
  '  #  
  
 # "   C
 
# 
-# 
    D,,

Figure 3.1.6

0
  

 
& 
 
#
& 
'&

  
    

 
    A 
  /

- 0 
   "   " '  

 
6   
  

 ! "   
  
   6   
 16
F

   ,
The Digraph of a Relation and the Transitive Closure

.      

 
  
" 
  
   
/


 0
    
DEFINITIONS

2
2

   # 
A   
   

     

 
        # 
A      

#  "      
   !
 # 
      

   
 E 
 !

  
#   "
 "   /
 0  #
"
!
 
  

  #   


#
! 

! /
 0  #

 
  

   
    "
 " 



2     
 
 
 #     
   
"

 !   

   "
 " 


     $ !  


  
 $ 

2

     # 

 
  #   
  # A 
/
 0  # 
 
  
4
5          5   

%

 /  0  #!    5 ,     %   74"

!  
" 
 #
  
  #   

 
" 
  
  
 #

2

%     
 

  #    
    
" 
 #  #  

        
 

Section 3.1

Basic Digraph Models and Properties

135

!

 /
 0!
5     
" 
  
 
  

- 
   

!  

-
   
 &
 "

 

 
  
   
  
  



EXAMPLES

2  

  #     5   "   " 
/ 0 / 0 / "0 /" 0 /" 0
   
    
  #
  
" 
 


 #    :

Figure 3.1.7 "

  
 
  / &
 

2 
  
    
   
 
 
 -"

 
    -#

6 # A 
    

-  
 # 6 !

   "   "  
 






  
          
 

          !     
  
"

6  
      

 - 
  
" 
    
 
A


    "   #  
 - 
  

Constructing the Transitive Closure of a Digraph: Warshall’s Algorithm

%  
 -"  
 # " ½  ¾         
 

Æ

!   +


$+
'(!  
4   
! &¼  &½     & !
 
 &¼ 5 ! & ½ 
 
  & !  5      !
  
 &  

" 
  &  
 &  
    
 & ½ 
  & ½

 /   0 /    

  & ½ 0 #"  
 
 
 
  & ½      ! 
" 
  
"

Figure 3.1.8 " 
 /   0  




 & ½ 

136

Chapter 3

Algorithm 3.1.2:

DIRECTED GRAPHS

1
&&* "
  / 2&
 314

2
 -"  
 & # " ½ ¾      
!2  
" 
   
 &
3
<  
 &¼    
 
   5  
   5  
3 /  0 

   
 & ½
  % 5  
3 /   0 

   
 & ½

 /   0  & ½ /    

 0
  
 & 
Activity-Scheduling Networks

3
   1!  
   
6 
 
  
 

 

 
   &  #
 
  
   
 

 
  
6   

  G     
6 
   "
 "  
 
 
6  
  
 

6   
  
 

# !
 

 
  
"
   
#   
    


 


   
6       


 
@  
 #  
6
   
 
 


Figure 3.1.9

  / )

 
' &
  

Scheduling the Matches in a Round-Robin Tournament

  
 
 
  
   
 
 
 
  

!
 # 

6  
  
  




  
 

  
 /
 
 
 
 

    
 
" 
0
3  

   @ !

 
1"   
"
 4


    

#

 + "# 

 
  
 ÜC'C
DEFINITIONS



2     
 -   
    # 
 




 
 




2  
  
  
"

        #  " 


#
  # 

Section 3.1



2



137

Basic Digraph Models and Properties

  


 
  5 / 0 

   


 
    

 #
 
 
"
      

 
@ 
 )
 
   
 
 
 <
    

ÜC



ÜC!

  
  

ÜC9

REMARK



2


    
 
 
  
-
  -   -


! #  "! 
 <   
  
6  " 
/
 C'0

ÜC'

 
  
      
 
 


   
 
-
  
 
 
 
   

'


Flows in Networks
 
 # 6   
  
 

   

 6       
# 6  
 7

 
   
!
    

      1
    
   
 

  


   

 
 ; #            
 # 6  
 1
 



   6     
 
     
   
  !    
 
# 6 
 
       
 

DEFINITIONS



2



 !  


 5 /  " " 0 
 
 # "-

 !
- !
 
" 

   " 2 



" 2 



2

) !

  

"   2 

  

!  
 5







( !


    

) 
 
A


¾

/0 5 ,

/  "  0 
 
 / 

# 

  
 
 0 #
 
" 

   " 2 

 

  " ! 

 
  " ! 

 



2




!

!



(!

#  <  !


#  <  

 !  

   
 ; # 
 
 

   
 - # 6       6   
  ; #  


  
  % 0
  ; #
 




   4
  ; #  /

      

  

 /



2



0

   !    A

   %   


  ; # # 6 
 
  

 
 /
" 
0 4
    /

0

Software Testing and the Chinese Postman Problem
  !



   #
F ; #  " # "
 



    
 
     

 !

 

3   #
!  # 



6  
  


     
  

 
 
 

DEFINITIONS



2

#



 




 
  

  #
6 
  



138

Chapter 3

DIRECTED GRAPHS



2     /  

0 

  #
6 
  

    

2

)"
  -#  


!

     " 

"   A
-#   
  


 -
&

2   #
F   ; #   


 
! # 

    

   "!  
 
  
!
 
  
 
 



   
      

     
  A
  4   #   
 " 6



 
 <   
  
   4"
   
 I 
 I 
! #


-#  4

REMARKS



2  
 
  
6      
 !    #

 <   
       /
    

 0 3 
 
! 

 
 
#  4
  
  
    





2 J 
 
 


 !  ; #  
  

 
F
  
 
    
 ! # 

   
 
  



2 7

  

 
!
 #
    

  !
   

 Ü9!
 "
  "     I 
 I 

 

   Ü9 
Lexical Scanners

     
   
 
 


   
 
#
   
   
! 

!
  < #  

    
 
  &  # +  #  

-  

#  
6    < #
    

 
"


" 
  

 
 

 


 
"
  

'
 
   
 


  
!
    , . " 
  
!  @  
  
 
"  
    
 
!  #     
 
  
  
"
 A
  "   ( 
! 
 
   
  


    
"
 A 7





 #
 6   
 


     


   
 
 3  A


   
  


  
 
!    
"
 A

Figure 3.1.10

  #
  5


 6


Section 3.1

139

Basic Digraph Models and Properties

3.1.3 Binary Trees
 A
!
   
        
 
  
 
 3 
! 
 
  
 3 

! 
 


   

B # 
#


 

Rooted Tree Terminology
DEFINITIONS



2

3
  ! 

 







2






   
     !
 /!    
  ,0


"


 ! 
    4 
      


    
  

  
      /#

4
  
    0





2

3 "

 





2



0!

 "








 "







2



  

 

 



 













/






  
      


"

   4 
      

  









/




 




3! 
 !

   



0





5

!

!



  








    #  
  
 "


 
A  



2



       


   


 



#     

 

"
! 
- -     "

#   



2



   


   # 
 " 

   # 
!

 
 
   
 



2



 
    






 

' 
) 
    

Figure 3.1.11


 
 / 0-
 

  

"





 





  


FACT



2

7" 
    






   





 "

  
 

140

Chapter 3

DIRECTED GRAPHS

Binary Search

  
  $    # A
 . A
    




!
        &      

 -
 

 

    6 !
    




  

 -



   

#  
  
DEFINITIONS



2      /0 

 ! 
  #  " 
 
6 !  
  6
  
 "   
 
  6
 
 "
 
   !
 
 
  6
 
 "       



2  
        " "!    "  

    @ 
   
EXAMPLE

2

    
 -
      
#    6 2
 H &  9     H C 9, 9'

Figure 3.1.12

'&
' 
)#
 
 
 '&


Algorithm 3.1.3:

! 
)#7
#"
 7


2

 -
  


  6 
!2
"     
 %/0 5      !
 8J%%       
 25 / 0
+
 / 5 8J%%0
 / 5 %/00
3  %/0
 25 *
"
 /0
7
  25 + "
 /0
 





    
 
  
  

 
   


 -
   "  
  #   
"
!     
 

     
   
  3    

!      

  # 
    "  # 
    B!  # -

 
   
 
 



 
 -
   ,/ * 0
    # 
 
"   
  
 
 


Section 3.1

141

Basic Digraph Models and Properties



  



 

6
!
     
 
 
  
  ,/ 0

References
$
)&&(  


  G )
! 

 

 
   
  !  7 !  -+
 ! &&&
$
),( K 
 -K
 ) )! 
!  
"  

   
!  -G
! %  ! ,,
$ %&'( ) 


 % %
6!
 # 
  7 !
 B

! %  L8# * 6! &&'
$ :C( 7






!  
  $ 
  ! I-B

! &:C

$)*&&( K % ) 
 K *

!
  
   !
&&&



I!

$ &( B 7  !     
 #



  
 
  
Æ
 
! 

! %! % 9' /& &0! HMH 
$ :'(   ! 
 %
 % ! I-B

! &:'
$ H9(   !    

! I-B

! &HC
$& ( B K 
 ! 

  !  6L
! && 
$#&( N 



  8  #
 !
  
  

! K 
+
 O  ! &&
$+
'(  +


!     

 
! &
 '  % & /&'0!
M
$+,(   +! 3    )
   !   7 ! ,,! I-B

!
/ 7  &&'0
$+&9( + % + ! .
  
2 

 
 
!  7 !  I! &&9

142

3.2

Chapter 3

DIRECTED GRAPHS

DIRECTED ACYCLIC GRAPHS
     

 7


 
 

   
 )
 I 
9  

 
 .<
 


Introduction
+
 
 
    
!   



 
 
! 
) +
 
 
 
  
    ! 
 4 



 "  
"


    
! " 

"       
    
6 


   ) 
"



3.2.1 Examples and Basic Facts
DEFINITIONS

2
2
2
2
2

  
     
  
  


#$ 

     
 
 

  
 
 
"    < 
 

 
 
"    < 

  
 
 

  "  
 "  " 

 
     "    
 

EXAMPLES



2 (
 )  
 
   1    
 

 
6 #

 
 
 P   
6    
   
  
   .



   
 1 

"   
 
6


   Ù
 Ú  
6 Ù    
   Ú 
     
!    ! 
   


  
 

   
 


! 




       

"   

 1 #



#
 
)! 
   #
  
!   1  
     2
" 
6    
 # 
 
"   
   "      


Section 3.2

Directed Acyclic Graphs

143

Figure 3.2.1

  

   
& /


2 $   $     / 
 !    ! 
 
 0 



 
    
  
 /
! <!
0
  

       
  3  
   
  4!
#



!
  

  

 
 
!
  
 

  
!
     /
  Ü 0

Figure 3.2.2 

  

)

2 

$'
  
   
    
  
!  
 # 
 " 6  %     &    
!

  

       
 
 " 6  
     
 &




 
 #
  

 ; # #   
  & 
)Q

8 !  
   
 
   
 "

#

     

 A
     ! # I   




 
   
 
 
)   ! 
  

/Ü 90

Figure 3.2.3 " &&
   




2 *    '   
 
  #    
  #
 

       
     9  #


 

144

Chapter 3

DIRECTED GRAPHS

# 3 
!   #

 
! 


   
 



 

    #
=
#  #>   =  
>  " 
 
 

6

  
 
  

 !
     


Figure 3.2.4 && 
%'

2      =

> 
 
! #  
   
 



"   
 #       #  
 
    
!


  

 
!   
   

 
  ! 6 #

 6 #
 B #"!  
#

)! 
   #
 
! " 
  # 
 
 
 "  
    


&

2 $ 

 %  "  
  
     !


 
   
 
    
!      



! #  

   A 
 
 

   
  

  "  #
6   
 = > 
   L
 



# 3 
 
 "  
"   A 
  / !
  -
- 0!
 
 

  
 
)

&

FACTS

2 7" ) 



   



  6
2 7" ) 

4 
! 

!   

  
2 7"  
 
) 
)
2  
" 
 
) 
)
2   
 
) 
 
 " #
6   


2   
 
) 
 
    
    "  
!  

1
 
!

 < 

 "  
 

 / 

  

Ü 9 A   0

2   
  
  
 
)   C  #
 

 
 
 

Figure 3.2.5

 
 
 

Section 3.2

2
2

145

Directed Acyclic Graphs

  
 
) 
 
         
 

  
   
  /

! #
6
 0 
 
 
 
    
  /

! #
6
 0

2
2
2

 )  "  
 ! 
    

 "
 )  


 
  



7"  
 #  
-
 
  "

 
 
 !
 
! 


  8

!


  "
 ½  ¾     
 

     
#        
REMARKS

2
2

    
  
   )!  $B
&9!

 '(
 $ :'! ÜM (

   
 
 
   
 



  ! 

 



 
 
   

 3
!   

   
#
 

E  Ü  3

 
!   

 

   
     

!    
 


    

    
!
  4   # 

 


3.2.2 Rooted Trees
3  
 
 
 
 & 
!  &  


)! 

  F " 
"
   
 B #"!   
  ) 
"
        
       !  $)*&&! Ü (
DEFINITIONS

2
2

    
 
 #  
 
 


    
  #
  " ! 

   !
 
   "  " !  4 
      
 
  
  

 
3 
#
   #    
6! 
 #
 



 
     

 
#

#
     3 
!  
  
#
  # 
- - !
    '!     " 
 

   

Figure 3.2.6

"% 

%) 
% 




146

Chapter 3

DIRECTED GRAPHS



2     
 


   

" 
   


# 
  
  # 

!   
! #   


 
 

   




2    
 - #

   "!  





  #
   

EXAMPLES

I"  7

  
     B
   



2   
  
  

 
  ! # 
   
   ! 

  
  


#  !
      

  
!  
 

 
   #
   :  # 
A #  " 

  -
- !   
 
 7
    
  #
 

6 1 $     3 # 
6 
    !
 A    
     A #  "

 
3   :  #       
"
/   "0 




 @      
   
 
 +
  

6 


  #  !   4   

    

@  
  ! 
     


   

    #  #

" 
 # 7

 '! #    # 
 
!
   
 # 
   


Figure 3.2.7

2

" 6
 % /    ##  


 '    
&/&

 
 
      
 1    A

A 
 
   
 #
     H  #
 

    



Section 3.2

147

Directed Acyclic Graphs

Figure 3.2.8

 
 


FACTS

2
2

7"   
)

  
 
   
 
  
 
   ! 


 " /  0 
   ,!


 
"   
DEFINITIONS FOR ROOTED TREES



2      
"    
     ! 
 ! 
      4  
       

2
2
2
2

 
 
     
  
"
3 /  0 
  ! 

G 
"  
 

 

3  
 
   "


  
   

2
2
2
2

    
      
 " ! 


    

  
" #   , / 
0
      
" 
   



 -   
    # " " 
 -  # 


   -   
 --
   # " 
" 



- 





"

  

"
    &

Figure 3.2.9

2& 
& 

) 8#
)9 




2    
    #     

 
 "

6
@



2     
  -
   #! " #
" 
 

 
!  
6
@ #  
  

 

148

Chapter 3

DIRECTED GRAPHS

REMARKS

2 !   !  !
 
  
6 A
 A  ! #  
 
     
  
  


 
  
   
! 
 
    
 
  # 


 !
  /9'C0  "   

 /9'C 5 C'90! 
 


 
  
    
 
2   ,  #   
   


 
 /
 ! 


0 B #"!
   !  5 
  5 !  
 #   

   !
 
 
 

 
!    
 @


!
 
  


@ 3  ! " " 
  
E    

 


Figure 3.2.10 

:   
  
FACTS

2  --
  

   - "

"
%
2 %  
 -" --
    
 

?


-   

-


#   

 
 
  

     

 
 
  
 
 --
 
Spanning Directed Trees

 "   
 


 ! "  
 


 
 3

!

   

 "! #
   
    B #"! 
 
!

   
 
 #
 
  


# #    
  #
  
   
    4   #
 
 


    
4  
#


   "     +
  
E 
,
 $)*&&! ( 3 
#



   
 
 E  '9 B # 

 # 6 

FACTS

2 3  
 & 


   
 !   A 
 

  #

A 
2   " " 
 
 &  

   
 
 " 
 
 &   
 

Section 3.2

Directed Acyclic Graphs

149

Functional Graphs




 

   !  "  
@  !

 


DEFINITION

2

    
 
 
  # 
 " 
   

EXAMPLES

2

  
   +  
A  
 .  
! A
 
 & # 
"   .
   # /  0 

 
 
 + / 0 5   A  
 !        "  .  B! & 
 

 /#
 
0



2 A

!     
      "  !   

 @           

   @ 
  
 ,     & 3  

   #    

Figure 3.2.11

"  &
 

'&  8
9

FACT



2 % & 
 

!

    
  

 
       
 

  
 3 &   
 

 
!
   "


    
    
 -

3.2.3 DAGs and Posets
 
" 
    # )
   7" ) 
 !
 "   
    )  "
#
    
 
 !  $ &,! :M(
DEFINITIONS

2

    

 
   
  
 


, - 2  


  !

E
 


2  


    ! 
 
  
! 
5 E

 
 2  


  $   ! 
 
   $ ! 
 $ 

150

Chapter 3

DIRECTED GRAPHS



2   !      / 5 / 0 

   



   !



    

2
2
2







$

7

7


 





/

   

   
 ! #

2



  

 ! 





!




 




5 

   
     / 5 / 0    
 # "
 
  

  
  
 

 

2  
    

 

 $  













  
 
    


 
 
  / 5 / 0   
 # " 

1
 
 
     "  





 

2  % 
    / 

 -
 
#    " 
 

 
 
  

1
 
 
#   
# 
EXAMPLE

2

%  5  9 C H , ,

     
    
 
  
 
 '

     


  

  B
 


  / 5 / 0

  #    


Figure 3.2.12

2
' & )

 
;

 
 

FACTS

2
2

3 

 
!   


  
 
   
)

7" B
 

 
)    

      / 


 #0

2

7" ) & 
    

#    
  /  
"   &!

    
 
  
 

  
3 
   ) &    / ! 



 
! 



 


 

"
! # 
     
6  
" 



&  &  
  
 
 /
  0 

  & 

3.2.4 Topological Sort and Optimization
3
)!  " 

#
    "
 




 
#     J   ! 
 <
   
 

 
"  

 

!  
 
6

 
    

Section 3.2

151

Directed Acyclic Graphs

"        
 
!
    <
 
 


 Æ 


    

  
DEFINITIONS



2       

 
 
 "   
"
 ½  ¾       



  
#-   -
"



2  
  !   ! 

 

 


  
 
 
 #  
  / 
  
 
!   
  
 
       
  0  
  

 

   #
 
  
 $ H9! Ü'(
FACTS

2
2

  
 



    
 
  
)

 

   
 
 
)
 A


  
   
Algorithm 3.2.1:

"& & 


 2
 
 &
! 2 

     & 
)E 

 #
0 25 &E % 5 

% & 
5 1

"   0  - 
 25
 "  0    < 
3   " !
2 &   
)
0 25 0   8# 0 
) 
 0 #

% 25 %?

, 

REMARK



2 
   
    # )
  !  #
  


  
  

  
 1
 #

 
      
  
! "   



  ! # 
   
 


   $)*&&!  : M :'(
Optimization


 
  
 
 

  
! #  
 
-# 



 ! #  
 
"     "
L      )!

    
 
  
"  

 
 
-



   
        

 

  
 




  
  
 
 $B%&C!  ,( 
 
 
 " 
   # "   
  

 
 

# A


 
  " A  
   
      
3 
 !       A!
         "
 "      3 
  !        

 
#
 "   "     

152

Chapter 3

Algorithm 3.2.2:

DIRECTED GRAPHS

!  )  0

   
 <
 

 2 ) & # "  ½ ¾          E # 
/0  "  / 0 
!
 
! 2  "
      
3
<  /½0
  % 5  
  / 0    # 
  / 0    ! %
Algorithm 3.2.3:

!  )  0

   7
<
 

 2 ) & # "
 #  /0  "  / 0 
!

 
! 2  "
      

3
<  /0  


0 25 &
  % 5  
 25
   0   0 
)
J
  / 0  

  # /  0 
    0 
0 25 0  

EXAMPLES

  
    
! 



 
# #
 
  ) 
" 


  

Figure 3.2.13

2


& /
 20-



.  $       ! # 
    # 


#
 2 

  "!
    !
    

6
 #    " 

 !  

6 "! 
6
 , 


 /  0 

  
6    
 
  
6 
 !
 
  
6
   
   



   "
B # 46

    "   
Q   
6       

Section 3.2

Directed Acyclic Graphs

153


  ! # 
  
/0 
      #   
"  
  
 

#  4 
 

# %

 / 0 5 
   
  
 / " # 0   
  
  
  
3
<
 2  /½0 5 /½ 0 5 ,! /8 2 ½ 5 
0
J
2  / 0 5 / 0 ? 
 /0 /   0 


3 
  
3
<
 2  

!  /0 5 /0!
J
2  

 
 /  0 

!  / 0 5 
 / 0  / 0?/ 0
  
!
 
   
#   /0! 
 !  / 0
    A  

  
"
A 
  
 
    
 
!  % $B%&C!  &(  

  
    
!    7

       
  
!  

2 
.  $  "  
  3 
  
6!
 
6 



  
    
    

  !  
  

   
   "       #  #
  1 
 
 
! #  # 
  

      #      %

 / 0 5 
   
  
 /   # 0   
  
  
  
3
<
 2  /½ 0 5 ,!
J
2  / 0 5 
 / 0 ? /  0 /   0 


3 
  
3
<
 2  

!  /0 5 ,!
J
2  

 
 /  0 

!
 / 0 5 
 / 0  / 0?/  0
  
!
 
   
#   /0

2 $
   +
      
 # # "


! # 
  

      #     Q 3




 # 6!
  #     
 
    

  /   
"
!   

 
  0!    
 

   
 
 / 

 ! 
# 

0 3  
 
)!

# 
6  
 " / 

 

 "       
 0!
  
  
 A    

 

# %
¼

 / 0 5 
      
 /   # 0   
  
  
  
3
<
 2  /½ 0 5 ,!
J
2  / 0 5  / 0 ? /   0 /   0 



154

Chapter 3

DIRECTED GRAPHS

3 
  
3
<
 2  / 0 5 ,!  /0 5     5 !
J
2  

  
 /  0 

!
 /0 5  /0  / 0?/  0
  
!
 
   
#   "
   / ¼0

2 +
      
 # # "! # 
  

      #    
 Q  
  
 
" 
 
    )! #

  
     
  7

 9 /

  #  # " # 0
2   B # 
  

  #
" 
 
"Q 3   
 
)!
  "
 

 !

 / 0 5     
   
  
  
  



3
<
 2  /½0 5 ! /½ 5 
0
J
2  / 0 5  /0 /   0 


3 
  
3
<
 2  /
0 5 !  /0 5 ,    5 
!
J
2  

  
 /  0 

!
 /0 5  /0 ?  / 0
  
!
 
   
#   "
   /0

2 %-
  +
    
 # # "   # 
   #   
 
  

 


 #   #
"Q   

 
-
 
 
 
 "
  

 
-

   3   9  
 
   ½      
  

"
  9 3     


  !
 
   #    # 

  
 


  !     
 #  #   " #
 
"

 
  

Figure 3.2.14 " ,         /&  


   ,       /& 

Section 3.2

155

Directed Acyclic Graphs

3   
 
)!
  "
 

 !


 / 0 5  
 "
    
     
  
  
  
3
<
 2  /½ 0 5 ,! /½ 5 
0
J
2  / 0 5 
 /0 /   0 /   0 


3 
  
3
<
 2  /
0 5 ,!  /0 5     5 
!
J
2  

  
 /  0 

!


 /0 5 
  /0  / 0 /  0 
  
!
 
   
#   "
   /0

2 %
-  +
    
 # # "   # 

   #    
  Q 

- 4   
"
 
 
 

 # 6!
 
   #    
 
"
 
 
  ! 
  # 6 
  
;   

  
 
   
    #  ;   

  #
 3
  9  
 
   ½      
  
 "
  '
 
  
 
    
  

    
 
 
  
 
  
   7

 : 
 !  
  !


 /0
 
<  ,
FACTS

2 
 M 
 
" 

  

 
  <
 
  
   
  ) /  


 "0
2 3  1 
  
  
  )!    
  
 
   
  
    
       

2  ) 

   
" 1   
  6 /1 

#   
 

1
 

  !

#   
 
1
 '



 60

References
$ &,( N 
! 



! B
  
 K "
 "! &&,
$)*&&( K % ) 
 K *

!
  
   !
&&&



I!

$B
&9(  B

 !
  
! I! &&9 /   
 !  
+
 ! &'&0
$B%&C(  B


 ) %
! 
  (
 )  
!  7 ! )
#-B

! &&C
$ :'(   ! 
 %
 % ! I-B

! &:'
$ H9(   !    

! I-B

! &HC

156

3.3

Chapter 3

DIRECTED GRAPHS

TOURNAMENTS
  
 
 
  

  
  
 A 
 7


  I
! 
!
 "
   
   4
 9 
" ! 
6 !    
 C N ! . !
 



 '  
 
 :  
 

 H G 


Introduction
 
  

 
  
 
   
 

 

  #  


  
 
  -   
 / 

0! 
- 
  !  
    

 ! 
1 
"  !  
  
!
  
  # 6 
 


 #
 "
 


 E   
!    
     
 

   






 
    K +  F 


  
 $ 'H(  
     
   
   &'H 3
 

 
   ;  
 # 6!  
   
  ;  
 
 C 
 
 4 "   " 


    

"



B #"!  "  
      

   

 

 
   
 / $B
8
'C(! $B
 ''(! $+:C(! $:&(! $H(!
$R &(! $)&C(! $
)&'(!
 $&'(0  # 6 
     
<
 
     
 / $
)&H(0        
  
!
# "  "
   

 "
  
<
 !    6 

 -K
 ) $
),(

3.3.1 Basic Definitions and Examples

 

   "
 "  #

   /
 0  







DEFINITIONS



2    
   
 
! !   


 # " 
   " /
 
0



2            " 
  #

 


   





   


2  "

 
     /   0 "  
/
 0 

    +
 

      /  
0 




#"

Section 3.3

157

Tournaments



2  " 
  
 "  " 
 
  


 
   " 
   
  "  " 
 
  






2   /   
0 
"  
 
     
" 
   
 3     
 /0 8  
    
  
 
   

     !   #

  
      #


    /0    /   
0 
" 
 
   
  " 
  
  3     / 0 /  / 00

2

  &  /    0 
 - 
    
  0! #       "  ! 

!


-
 /       



  








2   
     "- 
  
   #    
   
 " "    
 " "    
 
 
    
  
   '



2    
"

 
 &!   ,/
0!    

"


 
!
    
!   2 /
0!    

" 

 




3
 
 &!  - 
"

 

 
!   ( /
0 /  (  /
0  &   0

 




FACTS

2



 / ¾ 0 @

 - 
   
 

!
   
 
  

  !   "

   
 
"

     
 

2

$
C9(   / 0   -   /

0 - 
  "


  

  
"
"

  " 
 
   
  "!

/ 0


/ ¾ 0
/ 0

 
 / ¾ 0 S 5 
S




 A # "
  / 0
 " 







/ 0







9

C

'

:

H

&

9



C'

9C'

'HH,

&C '

,
&:

EXAMPLES

2

 
       9
 


     

,C'

158

Chapter 3

Figure 3.3.1

DIRECTED GRAPHS

"
  

  
  

2

)  

  <    
 
 - 
!


 
  



 3  "! 


   
  /
   '   / 0'
 
 

 0 3  ! 



 
  
 "

   

"   


 / $ 'H(0  -     
 

  
 
  -
    CC   
  !     
'
! 
  C 
Regular Tournaments
DEFINITIONS



2  
   
 
   #

 
  
 /!
 
     
  /0 5   

"    / 00   
 / 
 
0   
 
   # 
   /0   /0  5 





2   
   
 
  #


  "
1 
 
  
   " /!  
   %  
 ,/
0 
,/0 5 %!  

 
  "

    0



2 %  



      5 - ?  #  , %  
 --
     ,  
   "
    !
?  5 , 
 !  


 
   
   - -     
 
! #

  "



   ,     
 & # "-  /&0 5 

- /&0
A  2
 /
 0  /&0 
 
  
    &  


   
  #   
    #/ 0!  
#/ 0    
   

2

%  5  / 0   A A
 #  
! #  
!  
/ 
90!
 % 
   "  !

      
 

 


" 4
   /

  *
 
 0    
 

 
 #/ 0  


&     
FACTS

2
2

  
 
 
 #/ 0! #  5 ! 
 
 - 


$:&( 3  
 
- 
 - 
!     

  
/ 
90   "!  
 
- 
 /96 ? 0 M  
 


  
/96 ? 90  /96 ? 90 
 B  ?F
 F  
 BB
5 /96 ? 903
 B ? B 5 3! # 3    
 /
 B  


6#-B


 
0 $:(

Section 3.3

159

Tournaments

REMARK

2 4
!         
 
 

    
5 - ? 

#

 / 0  
6   ) !

EXAMPLES

2  &- 
  #        
  " " 
  
9!
  
 
   "!     
 
 

 5 )
  5  9 ' H

#

 / 0! #

0
1
8
2
7

3

6

5

4

Figure 3.3.2 "
 &

 & 


#

/ 9 ' H9

2   
 :- 
  #        4

   
-

 # / 0! #  5  /:0
  5   9 ." 
   
!
 

 
- 
  
  ,/
0  ,/0 5   

 
  "

    4

  :- 
        
    
 

    
 
   


Figure 3.3.3

/  99

#

Arc Reversals

 - 
 
  
  
  - 
 
4 
"
 

FACTS

2 $ ' ( 3


 
 # - 
 #  
   4! 

  
  
        
 
  4 
"
 
  - 



160

Chapter 3

DIRECTED GRAPHS

2

$: ( 3 
 
 # - 

 % 
A   # 
%
 !   
  
  
        
 
 
4  "
 
  %-



REMARK



2
  
 $HH(   
  
   
 #  

"- B
 
 A 
   - 
  
   # 
"
 
 
   
   
 B #"!

   
 

       
  
 / $H9(0
CONJECTURE

T
=
=* 2+
 $'9(
: 7"  -
"  
  



 
 #  "
   
   


3.3.2 Paths, Cycles, and Connectivity
I

  

 

-   

 
"  #


   
    
   
 
 "  
"  


 
 -K
 )   " $
)&'(
   6 $
),(

 



    
  
 A #
DEFINITIONS



2       /   
 0 
 
 & 

 
 


"  &      /   
0 
 
 & 
 
 
 


"  & /B

 
 

  

  
9C0



2   
 &   
/   
   0    " 
  
"

   &!  

  
 


    

EXAMPLE

2

 #  
     
  
  
 
  


!  

 
  
  #/ 9 ' H0 " 
,  '

C 9 H  : ,


 
   " 
   "

    




 !  

  









REMARK



2 
 :  
    



   

  
4
     A 
 
 "
"  
FACTS



2 $T 9( 7"  
  



 
 
   "! "  
  

    

 
 


Section 3.3

161

Tournaments



2  

#   

 4"
  
 - 
  2
/
0     
/0   

/0   



 
  
 $
C&(
/0   " "
 
   "   %!
%
!
  
 
 


  % $ 'H( /
 $B
 ''(0



2 $)
:(    
2    - 
  

4 

 
  
  4
  /  '0  
 

2

$  '(  
  

 - 
 

  

 
 

2

 
 ,/  0
   A


 
 
 
 
!
  
 ,/  0
   A


 
  
 
 
 /
$
),(
 $
&(00

2


 

$G ,( 7"
 
 
  - 
   
 

 
/ ? 0'  

Condensation and Transitive Tournaments
DEFINITIONS



2 3  
 
 # " 
          ! # 
  


 
  - 
   !      
    !     !   %- 
 # "-        
  #   
  #"

  "    


 
"     



2   
        

  "
! !
 $   !

 
 !
   
 $ ! 
 
 $ 
EXAMPLE

2

  &- 
      "-1  - 
  ! !
 !  # " "    
 " "  
 " "  !
 " "    
 " "     " 
    
A  C   / 0   / 0   / 0!
     
" - 
 #
"-   ! ! ! #   
 
  !
   
  
FACTS

2
2

$B
8
'C(   
    
 
  

"  


 

# A" 

 4"
  
 - 
  $ 'H(
  
/
0   
"
/0   
   

/0   

4 

 
 

/0  
   4 /,          0
/0  "   
 

             
   
  
 
 
 ! 
/!  
 
 $

( 0

162

Chapter 3

DIRECTED GRAPHS

2 7" / ½0- 
  


" - 
   
Cycles and Paths in Tournaments
FACTS

2 $
':( 7"
 
 
 - 
!
!    
 


  -!
-
 / $H,(    0

2 $K
:( 7"
 

   
 - 
!

  -! 9
-
 / $H,(    0




H!    
 



2 $
 :9(   "
 /
 0 
 
 - 
  ! #
  "   -!
$H,(    0

-





 !



 


 
 

-

 




:!

  /



2 $H,(   "
 /
 0 

   
 - 
  ! #
   "  
/
 $)G &:(0

-



-





 !



 


 
 

-

 





,!
 

Hamiltonian Cycles and Kelly’s Conjecture
CONJECTURE

>&&)* +
 / $ 'H(0 
- 
 
 - 
 
  
   /  0' ! 
  # 


 
  

EXAMPLE

2 
-   4

   
 
:- 
 #/  90 
 
   

,  



 
  
2

9 C ' ,E

,  9 ' 

C ,E

, 9  C  '

,

REMARK

2 N

F  1 
 
  # 6   
    7"
    1 
2     
& / 

!  $H(0E "
- 
!
C!  
 #
-1  

 
  
 $RH,(E  
 

   
 - 
  



  ',,,
-1  

 
  
 $H(   
 
  
    
   " 
 


#



FACTS


 ½
!  
 
  

-1  

 
  


2 $BU
& (   
 "  
 "!
- 
  







"

"

2 $HC( 7
  
 - 
   
  

 
  
  



    


     


Section 3.3

163

Tournaments

Higher Connectivity
DEFINITION

 &

2
 %  
/  %  
   0    " 
"  &! &   
   




 %    #

FACTS



2 $H,( 7"
 
-   
   
 


 
  

  "!   
   A

 -   
   " 
 
 "

  
9-   
  


 
 
  
 
  


 
 
    
   "!   
   A


 -   


2





$H:( 3  
%-   

 3 
   %    #

!  
-
   
   3  



 
  


2

$H9(  
 
 
 "
 % 
! ½  ¾      !
/ 0-   
  !  


 
  
    

½  ¾         
 





%

2

$ & (  
   -!
-
 ! " -  - 
  !
'!
 
 # "-1   
 
  -
  -! 
      
 4

   
 
 
 # /  90 / 
  
  

 - 5 ! # #
 

 

  $HC( 
 $
),,(0

2

$ ) %,( 3  
%-  - 
 #
"-1   
 
 

  "   




H

%

! 



 


%

Anti-Directed Paths

 
 , 
! 
 
"
 
       
 
  

  
   
 3

!     


  
 
  
   
 
   /

 

 

  
0!
  
    

 

  

  



3
 
!
 
 
  %    


%  
DEFINITION



2        /  0 
 
 & 
4 
 

 

   
   
 
  &       

 


    &
EXAMPLE

2

#
- 



-  

 


     9

164

Chapter 3

DIRECTED GRAPHS

Figure 3.3.4 "%  #


 
  #


)&
FACTS

2 $B
,,
( %  
 
 @    - 
!   
 C 
!
  4

   
 
 
 #/  90    
 " 
  
  

 
 
 / #
 A  "  
- 

 
 
  &: $):(0
2 $IH9( 3  "

'!  " - 
  


-


 
  
 / 
1  
   
 #
 A 

  

"
C,
$: (!
   
 #
  " 

"
H $ :9(0
2 $B
,,
( 7" - 
!
'H!  
 " 
  
/0



 
  
   
 /0 

 
  
 #   

 
 / #
 A  "  
½
$H'(0

3.3.3 Scores and Score Sequences

 ,!    


 
 B ) %

 $%
C (! 
  


 
  
    
 8


 <  

  


 /  " $&'(
 4 
 $)&&(0
FACTS

2 $%
C (  4 
 





  /            0! # 
 !     4    - 
 
 






 
        
  ! # 

!     4      - 
 
 




















%





 

%

5        



2 $B
8
'C(  4 





















%





 

%

5        











5

5



















/
 $B
 ''(0

2 %

 5 /          
  0 
4 
  
"  
# 

  
 

 !

 - 5        4 
  - 
 
 
  # 4

Section 3.3

165

Tournaments

½        

            



#

    -
 !     4    / 0- 

/ $:&(0

2 $"H,(    4

 5 /          
  0     4  


 - 
  
 
 
           
!
  
     4
 /,0! /! ! 0! /! ! ! 0!
 /! ! ! ! 0
/
 $&H(0

2 $VH9( %  5 /        0 
  4 7" - 
 #
  4  

4 

 
  
 
 



5 /  

   





   0

2 $*
HH(
 $*
H&( 7"  -    
"        
     

EXAMPLE

2 3  
  " 
  4 /! ! ! 9! 9! 90 
A      


,!
 !      4     '- 
 3 
!  '- 

   # "-1  - 
! 
 3 ! # " "    

" " 
 3 
   4 /! ! ! 9! 9! 90      A 
  4
  ! 
  
 
  '- 
 #   4 /! ! ! 9!
9! 90    
REMARKS

2  
" - 
   
- 
  #

   
   ! 
    # 4   
 - 

 

- 
  #

   
  
  ! 
   
 

- 
  # 4  ! 
    
     
 4       - 
 )"
 -     "
 ! 

  
  
 
 #     4 " 
 #
 

  $
:H(
2 )  &

 )"  
 
 -    !  # 


6  
6  
 


 6


 # J 

!     

 
6   
 
"      


   
6  
 


 
 
 /#    

  
 :   " 
 0     # 6 
  
  4! #  
 
   
"
 
 $
 9( 
6    "  


  !


 


 

# 
! #   # 
 
"
  #!   "
 
6              ! # 


F        -    
  
 . 
  
 A  -    "  "
  

   
6

  #
! 
    "
    
  $ 'H(     


 

166

Chapter 3

DIRECTED GRAPHS

The Second Neighborhood of a Vertex
DEFINITION


2


%

" 
 
 &    
 
!  
/
0!    

"  & 


  

-
   
-

 /
0 5 $



 ! (
( / 0 (  ( /
0


¾ 






(



FACTS

2
2

$&'( 7"  




 

"



 




 /
0
(  /
0



(



$B
,,( 3
 
   
  
!  




# " 
 
      
 '
CONJECTURE

7)
* 
 '


   +

   / $&'(02 7"  



"

$

  #





(

 

/$ 0




 /$ 0 

(

&

 -

3.3.4 Transitivity, Feedback Sets, Consistent Arcs
3
 
 
     

- 
  
/   
 
 -       
  
1  "  


 
  # 
  0!     
  

        1 # 
   
  /    
 #
   

       
 
F  
  


"0     

6   
  !  
  
 
  " 
 

" - 
      

  
  
   
      
 
   
 
 
 
 

DEFINITION



2  
   
 

 
     
   










  
 

EXAMPLE

2

%     
 # "-         #   
 %
#"  %!  
   
  ? !    5            
 /  ? 0 

 0 

6  
   !
  

  
 /  ? 0 

    


6  
   

Smallest Feedback Sets




 
6  
 - 
   4"
  A

"
- 
 / 

 0   
  / 0 5  / 0
    


    "  #
 
         
 


  


Section 3.3

167

Tournaments


  
  6 #

  $   " 

 "
"
 
  $ B+ &'( + 6    
  
-# 

6  
 

-#   
     $ )B+ &:(

$*,(
FACTS

2

 

  
 
 
  #  "



"  
 


 
6     / $
B3 &C(0

2

  
 


 
6  
 
   4
 
 
 


 
"
   
    / $
B3 &C(0

2

$
B3 &C( 3 # 


 
6  
 
  !  "
  #   
    - 
   

2

$
B3 &C(   
 & 
 
 
 
 
  




6      


Acyclic Subdigraphs and Transitive Sub-Tournaments
DEFINITION



2   
 
 
 & 
        

 

 
  &
FACTS



2 $:L:! H ( 3 */ 0   
     
 " - 

 

 
    */
 0
!  
  "  
 "½
 "¾ 

 ½¾ ¾ ? "½ ¿¾
*/ 0
¾½ ¾ ? "¾ ¿¾   "! "
  


 

#
/$'&! :(02
(



*/ 0





9

C

'

:

H

&

,





9

'

&



,

9

,

C

99-9'

2

$I
:,! 8&9! T
&9( 3 / 0   
     
 " - 
  


" - 
 #


 / 0 "! 




  9

:
9   H


/ 0 5

 C   9

:
'   H



¾ /' ':0
/ 0
 
¾  ?    

C9
/ 0

¾ / 'CC0 ? :  
CC



2 $ B
&H(
 $)) +&H(  
  
   4  5 /½  ¾
     ½  0! # ½

¿
  
 ½
 !  
 
 
 "         
     "   
  - 

      "
    "
 
"! !   
 
#" 

    /  0 /
 $,(  
   !
 
$
B
&(       
      -

  
  
 

 
0

168

Chapter 3

DIRECTED GRAPHS

Arc-Disjoint Cycles

 
 
"  
  ! / 0    
  
-1 

 
   !
 "/ 0     
 


 
6     
 !

 / 0 5 
/ 0
 "/ 0 5 
"/ 0! #  


 
6 "


- 
  

REMARK



2 8  
  4
 / 0 4
  
    -1  /0  
    
 
   ! # 
   #  $ )B:(
 4
/ ' 0/  0'
FACTS

2
2

 
  




! / 0
"/ 0

$N :'(  
,! / 0 ! "/ 0 
 !   

,!  

- 
   



 
6     
  
 
 

  

  
-1   
    /
 $:C(
    
$
),(
 $3&C(0


3.3.5 Kings, Oriented Trees, and Reachability
N 
  

    =  > "
  
 



   #  

# =6 > 
  / # 6   $%
C (0  
 


  
 $
H,( 
 

  
  7    

  # "
    
 
-
  
 "
"  
  =


 -> 
  
6

 

 = " ">      " 
 

 /   H 
#0   "!      6 


<
     
! 


 
-
  
 / ! 
    $&'(0
DEFINITIONS



2 


 
  
"

-
 
-
  
    



2   
 
 
"
-
 
-
    






 
   "  " ! 

 
   "  " !  



2   
  %  ! %
!  " " 
6
   
 %

  "        6 
FACTS

2

$G
C( 7"  
  

6  3 
! " "  

  
6 $%
C (

2
%

$
H,(    "   %
 !  
 - 
 # 


6  
 
 
%
! % 5 !
 /% 0 5 /9 90 /
 $H(0

Section 3.3

Tournaments

169

2 $H,(     ,


%
!  
 - 
 # 



6 ! 

 !
 

 " 

   6 
  
 

  

#     
 
A2 /0 % ?   
! /0  5 
 % 5 ! / 0
 5 % 5  5 9  % !
  ! ! /90 /  %  0     /  9  0! /C 9  ,0!
 /: '  0 / $ & (  
   0
%

2 $H9(  %-

  
  
"


 9% ? "   "!
 

#  

 4"
2 /0  
%-

 /9% ? 0 
 /0  
/9% ? 90  /9% ? 90 6#-  B


 

/ 0  
 
 
 /9% ? 0- 

2 $
H,( 3
 

 
 " " 
6  /
 $ 'H(0 3

!   
  "   %!
 

 

 %-

 $H9(
REMARKS

2    

   " 
  
1  
 - 


  
#   
   
    6     

 "
  #   $H(
 $+
H9( 

  
 
 )  #


"  )
 6 
 
  
 

- 
 #
  
$H,(    # 6  6    
  
     
 $&'(


2  A !
6     
  
   
     !
  

           

Tournaments Containing Oriented Trees


   
/   0 
 
 
  

!

   
/   0 
 -
 #


 "
CONJECTURE

7
* +
 / $+ H (02 7" /  0- 
  
 " 
   "    
EXAMPLE

2 

  H     9
  #     C     


    
 '- 


Figure 3.3.5 " 


  


REMARK

2 8  
    

 
    #

Æ   
  
 1!  
  


    4 
  
  = - 
 >
   4½ ½E
  
 /  0- 
 

  
"
     ."

170

Chapter 3

DIRECTED GRAPHS


 , 
 "

   


 
 
 /  
 $B
,,(0!

 #     1
FACTS

2 $B
,,( 3 + / 0   

   -  
 " -- 
  -


 " 
   "    !  + / 0
/:  C0' /7

 @ 

 + / 0
 $BU
&(
  + / 0

 ' $B
,,(0

2 $%+
I
,,( 7" - 
!
H,,!  


   - 
   
! #
     !

" 

  


    

" -  

2 $I,( 7
  
 
/ ? 0- 
  


    
 ?   # 

   
! 
  # 
 

2 $
&(   
   

 -


 -
 



-1 !    

A " ! 
 
    
   

 /
 0   !   
   /
 0  
 
 -
  
  
 -
  
  3   -
-  !    " 
  "


!  
 -
  




 1  -
  
 
Arc-Colorings and Monochromatic Paths
CONJECTURE

0 
? +
 / $

+ H(02   
  "   %!  



 "   /%0  
 "
-
  
 "
" % 
  

   /%0 " #    
   " "      !  
   
 
       "   
REMARK

2   1   


   
 "
   
 

 "  
  



 
 
! /0 5  
 C' 
# 


 /0 5   
 
   &- 
 
  
 
  

- 
 #
- 
  # 
 / 0  / $

+ H(0 3 

! 7U

6  / 0 5  3    " 6 # 
 /%0  A   %
     
  1  
 
# /
 $%
&'(
 $,,(  


   
 

 "
 4 0
FACTS

2 $

+ H( 3 
 
 
 
 
 # # 
!  

"
   
   " "  5
  !  
   


    
 / $H9(  
   0
2 $HH( 3 
 
 
 
 
 #  

   

   

- 
 

" - 
   # 
 



!   
"
   
   " "  5
  !  
   
 
    


Section 3.3

171

Tournaments

3.3.6 Domination
3    
  
" 

  
 
   "
 
 
    B #"! 
 
   
     



   

!   
    

  
 
 
#  
   5 / 0 5 %  
"   % 
 
  



 
 6 #!    #
  "!   
 "
  6 # 

  

"
  %  
   
/0 
    Ü&
DEFINITIONS



2    
 
 
  

"       
    "   



 "    
 "



2    
 
 
  
  
  
 

 
 "-
  !


1
     #" 
  
 

   



2  
  
  /0 
 #    
 #


"    
  "!
 
 




2      
 
    

 
 
    !   5 / 0
EXAMPLE

2

%     
" - 
 # "-         # 
 
  #"   "
 
 /  0 

  - 
  
# "  
 
 " "

-
!
  
 
 "       /  0
"
-
 /   " 0  !  
6      
  
!
 " "     
    !   
 
   

5 / 0 5 
FACTS

2

$):(  
 "   %!
     

   


 %¾¾ ¾! #   /  90   
     4

 
- 
  
 
 %  /
 $BB,(0

2

$%&H(   
  
 
 
  
6 
 
 #  #  
 "! 
   


 3 

! 
 
  
 
 - 
 

      !
 6
  
 #  #  
 "    
  
     


2

$%&&(    
    
  
 
 
 
 
   
6   
!

! 



 #   
  "

 
 



  "    
1
       
 
 
 
"
   
  
 # 

<  $K%&H(

2

$<<'C( 3  
 - 
 #

A 5 / 0
*¾  *¾ *¾ ? 




!    
   



172

Chapter 3

DIRECTED GRAPHS

3.3.7 Tournament Matrices
  

# 6   
 
 
  
  

 )
  K +  
 8 K I


 /
 
 
   Ü  E 
 $):(0E
# 6  B K   $ '9(   
 
 # "  #
 
  

<    =>! !    F
 "  
 

E

# 6    
6  $'C( 
    
 
 # 
 #  
 <
 
<       

 V %
$%H
(     
 
   # 6   
 
6 

   
    
 
  
6  <
 "       " 
 <   
 
   
  #
6 

DEFINITION



2      
4
 
 6 5 /- 0  ,F
 F! # ,F
  
 


 - ? - 5 !  

 
  


 
"    "! ½  ¾      ! 
 
  !
     6 5 $- (     ,- 
 " 
- 5



   
 
, #

!
 
 
  
1
 
     
  
"
   "
EXAMPLE

2

  
 
   '   #     '

Figure 3.3.6


 
,

REMARK



2  

 "
 
 
   
 
 6  
 6 ?
6  ? 2 5 7 ! # 2     
! 6    
  
 
6 !
 7    
 

F   "!

1
 
 6   

 
  
 
1
 
 (        
"   
 ( ! !  

  
 /  
 6 5 / ½( / 
!   "

  
  

   
 
    


  


Section 3.3

Tournaments

173

FACTS

2 $)'H( % ½  ¾           "
 
   


 ! # ½

  
   , ! ½
/  0'!
 
 /  0' ½¾ !  5          "!  6 
   
 

    
  2 
  "
  6 !     ! 
 2
6
   
 
 

    
  
 

2 $
)NI
&(  





! 
 
   
 





    "
E 

 
 

 
 "
 
 
  
B


  
 
 /! 6  6 5 2 0 

 
   
 
 # 

  "

6

2 $&C( 3  
   
 
!   
6    4

/  0' 
 
  
  5 , 74
 
 
  


  A
 " /  0' /#        
 0 7

  

" 
6 /  0'     /  90   A
  

 ! #  "
/  0'9! 
  
    
 
  
 
   /  0


 /  0  #  ,F /
 !
 /  0 
 

F 
"  
 ,   /    0   0 

# 
   #  /  0  " F

,   /  0    /
 !
  
  ,F0
2 $&(   
 
   
    
 -    


 "       
  - 
 

 
 


<
"   4
   
 
 
   
 =Æ


>  
   

 
 
   
 
 

 
  4
 /'0/  0!
 4
 
 
 

-   

  "       
  - 


3.3.8 Voting
+ 6 
 
  
   
! 
 
" - 
! 
   "       "

      
 "  
 
 A
     

   


  
 
 

 
 
 "    ! 

 

  



   
  
 
  +! 
  
 
 ! ,$


!
 ) -
   , 

 

Deciding Who Won

 
  "      6
=>

" /    

"0 " 
 "   
" 

  =>

" 
 

"  



 
     
  =#> "

 
 
  
"     

 7
     - !
"

    !
 4
   #      
 
     
 
2  " 
    / 


 0! " 
   
  "
  
1
 
  
 
! "
 
 #

6 "  !    # 

"  
 
  =
>   "  
 /

   

0! " 

  " 
" 

 = " > 
 ! " 


174

Chapter 3

DIRECTED GRAPHS

 
  


" - 
 / 
6 0!
 "

 


 
  
  /  
 4
 0 

  

 K- %

F   
 $%
&:(
DEFINITIONS



2      


 / 

"0
 
   "
 

 / 

"0 !
  "  

1    " 



2    
 & 
  - 
!

#  
 "  ! 
 "-  !
 "
 
 "   & 
 

 

 

1    - 




2   
 &    
  "   &   
1   
 




  

     "  &! 

   
 "  /


      "    

" 

  "
 & @ "    
"  


 0



2      
  "  
      /! 
   
6    

 


 0!  



   " F   
1  
 
 
   
/!  
   
1   
0
REMARK



2 3  A  

1   
!     "  
   


A   =

">!
 
 - 
 
    
 

#   

" 
=" >  !  
 
1   

 " F   
1  "   3 

   
"   
  !   
1   
 
 

EXAMPLE



2    : 


  
1   
       
"
C- 
 ½ ! ¾ !
 ¿  3  
#   ½ ! ¾ !
 ¿ ! 

 
 ##

  
 
 
 "  
 

 " 
#   


!      
   
! " "  
 

" 
 !
 "    
  

" ! !
 "     

  
1   
 


    

    
!   # 


1   "     !

1     "!
 !

1   "
 

Figure 3.3.7

" +
)

   


Section 3.3

Tournaments

175

Tournaments That Are Majority Digraphs
FACTS

2

$C&( 7" - 
 /! "  
   0   
1   
    

   ?  
!   !
  ?  
!
  "

 
% -/ 0    

    

 -"  
 

 


1   
 


   -/ 0  # " !

 */ 0
   

    

 - 
 
  


1 
 
 


   */ 0  # " 

2

 
  ! -/ 0
/CC ' * 0 $C&(!
  
 
 "  

-/ 0 ! /" ' * 0 $7 '9(

2

$ 'H(    */ 0 
#
 ! */ 0 5 */90 5 */C0 5 ! */ ? 0
*/ 0 ? !
 -/ 0
*/ 0



2

$
&&( 3  
   
    
1   
!  
 ! ' !
!  
  



 - 
  !  
   "


  8  
"  
   



  !  

 /  0
   
       8  #  
  
 
  3  ! 
    ( 
    
 

   
" - 






Agendas
DEFINITIONS



2  
  
 
 

" /!
 
   "


1   
0



2      "  
4
"     #!
"


/½       0 

"!

"   

   
A " !   #  

      " !   # 


     " ! 



2 )"

1  - 
 



/      0 

"
"   "   !      

" "" 

"  /!  /  0 " 0 

    "   
1  " 
 
 
 3 
   


        
 !
     !  
    !     
   

    "  
6  
    
 /      0
EXAMPLE

2

)" 

/  "  0
  
1   
  #   
 H!

"               

"  
  #

176

Chapter 3

Figure 3.3.8

DIRECTED GRAPHS

-+
) 
  
 

   


FACT



2 $::(  
  
  !    " 
 
  

 
   
    

   "   

        / 
   !  $&
(0
Division Trees and Sophisticated Decisions
DEFINITIONS



2 )"


/½  ¾    0!     /½  ¾     0  


! 

! 
 !        "

   - 4  

/½  ¾     0E    

 /½  ¾     0E
!  
,

 !
"

"
 # 

 
4  /½  ¾    0!

/½  ¾ ¿       0!  
 

# "

"
 ? ! 


/½  ¿       0!
 

 /¾ ¿       0



2 %  

1  - 


 /½  ¾      0 




" "   "          


 
      "   
"  /½  ¾     0
  ! # 

  
 
 "

"
    "     
1 
    #  #

" 
 
6    


 

"   "  E
 "
!   ,
 !  ! 

  
 
 "  
"
   "     
1      # 

  
  # "

"
 ?  

  
  
FACTS

2

$
HC(    " 
 
  
 
  

   
 "    
   
"   

 4

   "   

 
  


" - 
 


2

8

"  
 
    
 "   
/."  $::(
 $H,(
  "  $&(0

2

$&:(   
 



  #   "   
   
 "   
 

 
  
         
- 
 

!
  

!    

 



EXAMPLES

2

 "    

/
  $ 0   #     & )" 

1   
  #! 

  

"
 
 ,   " 

Section 3.3

177

Tournaments


 
   "

 

  
     ! 
    
   
"    



 8  
 
    $ ! # 


 
 : 

Figure 3.3.9 

/   
 
+
) 


2  
1  9- 
  #     , 


   " 


  
 :    ! 

  
 
   

/  
0!    
   
  
   

Figure 3.3.10 +
) #

Inductively Determining the Sophisticated Decision

 

# 
 


      
   

    
  #
 
   A  /


 2 /$ 0   
-  " $ 0
FACT

2 $+H9( %  

1  - 


 /½      0  


    

" 
 
6   "    3"
A
 4 $  $     $
 

#2 $ 5 !
   
 ! !


 
$ 5
 $

 





 
#

2 /$ 0

 $    
  

References
T
T
! I 
! MH   
'
   " 
!  !
$'9(
 T
$
  ! < 
  I
! &'9

178

Chapter 3

DIRECTED GRAPHS

$
':(  

! 
  

    
  
! ! %! / ! ,
/&':0! H MHC
$
:H(  


 N  !   4   

  
!
&!
  
 /&:H0! 9M9&
$
 :9(  

! N  
 I  

!  
 
   
!
&! 
!  
/ : /&:90! MH
$

H(  


  
!        9- 
 
!
  
 %!  /&H0!  M&
$ B
&H(  
  B
 !   " 
  
 #  
  

   ! 
 
! 9& /&&H0! HCM&
$"H,( I " !
C:M'9

  
  4   
! &!


  

$
HC( K  
6!  
 "   


 
!
 0 '
 /&HC0! &CM ,'

9 /&H,0!

$ 

$
&( K 
 -K! 7 -1  -
 -
    

 


  
! &! 
!  
/ C /&&0! M 
$
B
&( N  

! % + 6
  B

 ! #  
  

 
! G"
3
 &!
  
1 /&&0! H M&9
$
),,( K 
 -K! * ) !  * ! 

  
  
 
"   
! 
 %! 9 /,,,0! ::MH:
$
),( K 
 -K
 ) )!
!  ! ,,


  
"  

"   

$
)&'( K 
 -K
 ) )! I
! !
  
   
!

 

! C /&&'0!  M:,



$
)&H( K 
 -K
 ) )! )
<
    
2
" !

  
H /&&H0! :M,

&!

$
B3 &C( K-I 
T
T ! . B ! ) 3

6!    !  
! 
"  
 
! 
  ! %! /&&C0! &M:'
$:( K   ! .
W 
  F     
 

 

 

! %! '
! $! 
  : /&:0! CMC
$:C( K-   !     
 
 
! 'CMH,  +
2 $ ! 8  B

! &:C



$H( % + 6!        
  

 
  
2
 

" " ! 9MCC  

E #

! &H! 3 %!
$! 3 
  C!
  J" I!
 ! &H

Section 3.3

179

Tournaments

$N :'( K-   
 * N 
 @! J   
 


F 

" F   ! )! ! ! )! (! '
4  
4 , /&:'0! H M&
$H( K-   

 
C /&H0 M9 

  
!


   
2
" !

&!


$+:C( % + 6
  K +
 !  "   
   
!
M9H  )     
  
! 
! $
! 
 1567! 

!
&:C
$)'H(  

 3  ) ! .  

     
 
!
/ ! 

! %! $! :9 /&'H0!  M C
$):(  

 3  ) !   
6   
 
!
  
  ! C /&:0! M H
$%H
(   

 V %! J       
!
 
/ /&H 0! 'M::

3 

&! 
!

$VH9(   

 V %!  
  
   
 #  

  " ! &MC  

 
  
! 
 I! &H9
$,(   

 K ! %

F 4
    
  

    
   
" - 
! &!
  
H /,,0!
99MC9
$
C&( I
 !    

    
  
! ! )! !
$! I
 9& /&C&0! CMC
$
)NI
&(  
!   )  !  K N6
! 8 K I


! K 

! 

 

    "
 
 
 
! 3 
  
  ! '& /&&0! :&M& 
$ )B:( ) 

!  )


  B! )
 #    
! &! 
!  
/ , /&:0! M9
$ ) %,( ) !  K ) 

 B %! I
  " 
 
 
  
! &! 
!  
/ H /,,0!  M,
$ )B+ &:( 3 
 - !  )T ! . B
  + 
! 8#

    
   
 ! 
 %! 'CL'' /&&:0!  &M
C 
$ B+ &'( 3 
 - ! . B
  + 
! . 
  
 
    ! %! '
! $! 
   /&&'0!  MC'
$
C9(  % 
"!    
 
 !
 M9,

/ ! %! / ! ' /&C90!

$7 '9( I 7U 
 %  ! .  
    

   
 !  ! %! ! ! ! $! & /&'90! CM 

180

Chapter 3

DIRECTED GRAPHS

$H ( + 
< 
G
! .  
 

 
   



   
! &! 
!  
/ /&H 0! HM 
$&'(   ! 4

 
2    
F  1! &!
  
 /&&'0! 9 M9H
$%&H(   ! K  % !  <
 N  !   
 
    
 
 
! &!
  
& /&&H0! , M,
$%&&(   ! K  % !  <!
 N  !   
  
   
! &! 
! %!  
! 
!  /&&&0!
'&M:'
$H:( I 

   
! B

 
  

"  
 
 
!
  
! /&H:0!  &MC,
$'C(   
6 ! J       
! ! &! %! : /&'C0!
&C:M&'&
$)
:(   )
 ! . 
  
 

 

 B

 

! &! 
!  
/  /&:0! ''M'&
$):(  )U

! - B

 
 
   
! &! 
!
 
/  /&:0! 9&MC:
$)&&( K  ) 
 N  ! %

F   "! 
  &!

! , /&&&0! &M9
$):(  % )


 K B !   " 
  
 

 
! ! %! / ! 9 /&:0! 9CM9H
$)&C( ) )! 

 
   
 

  
!  !

2
" ! &!
  
& /&&C0! 9HMC,C
$)) +&H(  )
!  ) T
T
!   
T
 I +
! -
 -
"  
! &! 
!  
/ : /&&H0! HM&'
$)G &:( * )
 % G
6
!  
   
! 
  ! %! :&
/&&:0! :M C
$B
,,
(  B
"! . B

 
  
   
! &! 
!  
/
/,,,0! M 
$BU
& (  BU
6"! B

 
  
   
! 
! 
! 
!
 /&& 0! CM 
$B
 ''(  B


 %  !          
! 

!
%! % : /&''0!  M9'
$B
8
'C(  B

 !  R 8 
! 
# ! $

 %   

    
' 
 
! K  +
 O  ! &'C

Section 3.3

181

Tournaments

$B
,,
(  B
"
   
T! . B

 
 
   
2
    
F  1! &! 
!  
/ :H /,,,0! 9 M: 
$B
,,(  B
"
   
T! 
    
2

  
       

 F  1! &!
  
C /,,,0!
99MC'
$BU
&(  BU
6"
   
 !    
! 

 
/&&0!  M ,
$BU
&:(  BU
6"
   
 ! . B

 
  
  

! &M C  

"  

"  
  8

 " 1559:!
  J" I! &&:
$3&C( ) 3

6!  

 
6
 ! 7
  &! 
!  /&&C0! &
$K
:( .  K
6 !
&:



 
   
!  ! J"  
!

$K%&H( ) K<
 K  % !  
 # 
    
  
! 
 

!   /&&H0!  M 
$%
C ( B ) %

! .  
 
 
   

  333
     
  ! / ! %! / ! C /&C 0! 9 M9H
$%
&:( K- %

! 

 $   %.
 ;!  ! &&:
$%
&'( G %6
  
!     
   -
  
! 


! 99 /&&'0! CMH
$%+
I
,,( X %! -+ +

 K I
!   
    
!

  

 ' /,,,0! 9M9:
$
H,(  
!  6 6  ! %! %! C /&H,0! ':MH,
$
&&( K 

! . -
1  "  

 ! %! $! $! : /&&&0! &M99
$
&( * 
 
6! 

-
   A B

 
  
   
! 
  ! %! ' /&&0! &&M,
$::( 8  

! )
- 


      "  ! 

! &!
 ! $!  /&::0! :'&MH, 
$H,( 8  

!  # 
      

 
1  "  2 

- 


      "  ! 

! &!  ! $! 9 /&H,0!
'HM&'
$&C(   

!  
6   
 
! 

! %! % ,
/&&C0! ' :M' &
$ 'H( K +  !   

! B
! 
!
 + ! &'H

182

Chapter 3

DIRECTED GRAPHS

$  '( K +  
 %  ! 
 

 

 
!
%! / ! C /&'0! 'M'C

!

$8&9( G 8
 %

!      
   
 I
6   
!

  
! , /&&90! ' M ''
$I
:,( 7  I
6
 N  !   
 1  7U

   
 
! &! 
!  
& /&:,0! CM H
$I,( G I "!
C&'


#   
 
 
!
  
! /,,0! C&M

$IH9( G I "!  B

 
    
! C&M'& 

 
"  $" 15<9! J" 8 " 
! 8 " 
! &H9
$,,( N  !     


 
-
  
!


! 9' /,,,0!  M9
$T 9( % T! 7 6 
  
<!
/& 90! &M9 


 

 3


 $ 

! < 

:

$'&( N  ! .  
  
   
 
 
! ! %!
/ !  /&'&0! 'M':
$: ( N  ! 74"
  - 
 "
6-
 "
!
' /&: 0! ' MH,


 %!

$H,( N  !  
 #    6 
 !

! & /&H,0! H,&MH'
$H( N  ! 7" "
6 ! 


%!




H /&H0! & M&H

$H9( N  !    
 


 !  

  

 


"
   
! $

;
 3 
   %
 ,: /&H90!
M
$HC( N  ! #  

  
  # -   
! :M
    
! "
 C  
   %! $!! 8 -B

! &HC
$&
( N  ! 
1   
2 
  
 "   

  ! %! $! $!  /&&0! M&
$&( N  !  
  # #
     "
 
 
1  "  #


! 
  ! %!  /&&0!  MH
$&'( N  !  
2  ! 6 ! 
<
 
 
 !

 

! C /&&'0! :M



$&:( N  ! 74



2

  

  
 "   ! $   0 '
9 /&&:0! ' M :H
$:&( N  
 % + 6!  
! '&M,9 

  
! 
 I! %  ! &:&

$    

Section 3.3

183

Tournaments

$:( N  
 7  #!  
 
  

 4"
  6#
B


 
! &! 
!  
 /&:0! M H
$H9( N  
   
! 

  6    
! :MH
 

 
  
! 
 I! &H9
$BB,( N  !   
!   B
   B!
 
 
 
   
! ! ,,
$+ H ( N  
 8 + 
! 7  -   
!
$ $! %! 
 H /&H 0! ::M H:
$ :9(   
! - B

 
    
!
 
/ ' /&:90!  9M9

&! 
!

$ '9( B K  ! 
  < 
    
 


! , M9
 )      %
-  
! J" +  I! &'9
$T
&9(  T
<-
! .  

 
  
"  
!

  
! , /&&90! ':M :'
$

+ H(  
! 8 

  +  #! .    
 
   
  
! &! 
!  
/ 9C /&H0! ,HM
$HH(!  ! .    
 
  -
  
!
 
/ /&HH0! ,HM
$&(  % 
! .  
 
!
CM 'H

3 
  
  !

&! 
!

'L'9 /&&0!

$+H9( N  

  +
! J " 
  
 " 
  # 

   

 ! 

! &!  ! $! H /&H90!
9&M:9
$*,( 3 
  /3 
 - 0
 . * /. B 0!  


      
"   
 

   #   

/
0! 
!  !  ! (
! H /,,0! : M&
$ & ( R    ! 

  
 


    
!
 
/ /&& 0! HMC

&! 
!

$:L:( K ! .

6   
!  
  /&:L:0!  CM H
$C&(  
!  "   
! 

!

%! %

'' /&C&0! :'M:' 

$<<'C( 7 <6
 ) <6! .
 
  U

 7U !
9& /&'C0! &,M& 
$&H( I 
!  

<
   4  
!
/&&H0! C:MC&
$: (

%! =!

&! 
!  
/ :

  
!  B

 
 
 
   
!
, /&: 0!  M H

%! !

184

Chapter 3

DIRECTED GRAPHS

$H,(   
! B

 
-   
! &! 
!  
/ H
/&H,0! 9M' 
$H(   
! 7 -1  B

 
 

  
   
! 
!
3 %! $! 9C /&H0! CM'H
$H9(   
! "   
! ,CM  
  


!
 ! &H ! 
 I! &H9



$HC(   
! B

 
    
  
!   

%! : /&HC0! C&M'
$H'(   
 ! I

  
   
! 
! 

! %! $! &'
/&H'0! ':MH,
$HH(   
!  "
   
! 
 %! : /&HH0! : MH'
$G
C( B 7 G
 
! . #

-  
 


   !
+ &! %!  /&C0! 9,:M9
$G ,( % G
6
! 
  

  
2 

  
! 

%! 9C /,,0! &MC 
$+
H9( N +

! )  6 
! 
 

! 9C /&H90! M
H
$*
HH(  X *
! F  1       
 /

/&HH0! 9HM9H9

0! > -

$*
H&(  X *
! .   1        
!   $! / !
9 /&H&0! H,9MH,H
$RH,(  V R
 ! 7"  
  
 
 #
-1  B

 
  
!
&! ?' 
   " $   (
! ) ! /&H,0! :,MH
$R &( N- R

 R-   !

 $! : /&&0 HHMC


   
 M
" ! &! . @!"

185

Chapter 3 Glossary

GLOSSARY FOR CHAPTER 3

 / ) 


2
 
 
  
 #
  
6  



 
 

   1

 / )  @


2
 
 
  
 #    
6 

 

 
 

   1


+) 
,

M 
 
2 

 
  

 






 

 


  # 



  

M  "  2
 
 

" /!
 
   "



1   
0

&
 &



 




/ 

 
02




  

M   "  2
4
"     #! "


½       


/

0 

"!

"

" !   #  










 





  A

    " !   # 

   " ! 


  /
, 




M 
  2
" 

   4 
  

   E 


 #


)&

 

M 
 



 
 

 #




&

M 
 



 
 

&

&2

4 
 
  
 
  

&

      

 
 
   

&2

4 
 
  

  

&

      

 
 
   

 )

&  #

@

 
 


#
'   



2   #!  



 !



#



#
!



5 

2 "  

2 "  8 

2 



2        -
2


  "  
 "  " 
 


     "    
 


'
' 
) 


 
 

2      

2
    # 
 " 

   # 
!
 



   
 
!

'&











2

   
   " "!    "  



    @ 
   

' 
)#
 


/02

 ! 
  #  " 
 
6 ! 


  6
  
 "

  

!

 &
 /
, 


!



 
 
  6
 
 "  

 
 
  6
 
 "      
M 
  2

"  #  
    



&



E



M 

 2

 #   
    
! # 

   
  



' & )


 

  



M 
  /




 02   

 


 


# " 



 


186

Chapter 3


'& &
 


 02





  # " 
" 
 



-

M 
  /

& -#
) 


2


--





 

DIRECTED GRAPHS

 
 







"

  

"


&



2

  
  
 # 
 
   "! 



     


 


 #      
 ½  ¾      2
 

½  ¾         
 /   0   / 0 
 

 

   
   
"      
"    
 

    
  2
 
   #  "-  ½ ¾     
   
" 
  ½  ¾         / 0! # 
  


 
  - 
   !
  # "
  

 #"

  "    


  "     


M 
 




# "-

5

2

 

,

2   
 
  "  
     -

 /! 
   
6   





 


 0!  




   " F   
1  
 
 
   
/!  
   
1   
0

2

 % 

M  "  2

 

 / 

"0

 / )





 / 

"0

!
 

"



 
   "



1    " 

# 

0 M 
 - 
  
2   <
  
       
     "

/  " 
 
 -"  
   0    /0
      "-
 
 
 


/ 


" 
 



/0



&

    


M 
 


 A% %
.
/



2 

&2

 
 
 

 
  


 &

 02

M 
  /



 
 # " 

 


 



1
 
 
     "  

/


M 
  /

 


$

 02  
     


 
 $  

202
 & 0 -

2

  

 



I
  
2
    
  
 # 
6 
" A


   6 #  #
  1   


     8
 
!

 
-
 
"
    A 
  
  

  A

$

   


2

 
 




M  
      
" 
1 




/

½  ¾     

 
    !   

  " 




/ 




6  
   

0

M 
"

- 

!

0 

"2  
 !    

 
  2


 # 
½  ¾      0

 





 /


  /!  
0  

4  
      



   /
, 

      

M 
  2
"

E 

"



 




  
 
  

  

   4

Chapter 3 Glossary

187



2
 

! )& 2
 
 #    
! !

 
 
!
.-/
! 
2
 
 #  
 
      &
  
 

!


   
&  # 
A  2

  
 #  "
     
   !
 # 
       

  
 

%.2      &    

! %.&) 
2
 
 #  
 
   E    
!

    





)& 
2
 
 #    



2  0 0
'&2  A
-#   
   
" #   





)&2

  





 /  02
   !   #      

  
!
 #       

   3

 
# ! 
 #
   #
  





2

 
 
  

 
  #

 
 
  
  





2
 
 #  
 
 




%&. M   ¼  2


 4 ¼ ½ ½ ¾  

½   
 "

!  
  / 0 5  ½

 / 0 5  !    5       

 


¼ -  #
6


/   
 M  "  2    ! A  :

    M 
 
  2
   "    
 " "
      
    "   


   /  $

0M
"  
 
2
  

"


  

  
 
!


 M 
 
  2
  
  
 
  
 "-

!

'
M 
 
2   

 
 
    E

  !


1
     #" 
  
 
    
  5 / 0


'&)#
 &

2  
  

# / ) M 
 -"
 
2   < 
    

 
        
    /0  /0
/  0 M 
 
   
2

  #  
 

 
 
 
     
     
 

 

! 
 

!
  


#

&
 

 
 2

  #
6 
  

 # 

'.   
 M 
 
  2
  
  
   
   
 
   


A% %
.2   &

188

Chapter 3


 

 M 
" 
   
#  
    

DIRECTED GRAPHS


 
2

 #  

 



 &
2
 
  # 
 " 
   
 &  )& /  

0 M 
 
 &2
 

" 






 




&

 &   /  

0 M 
 
 &2


" 

 
 
 


&

;

 M 
 2


 -
 
#    " 
  
 


 
  

1
 
 
#   
# 


2  
 

  M 
  2 
  

  
     
#'
  / 
0 M 
 
2
  
  #






"

#
 M 
"
/! 



E




 


 

 / 0



/ 

# M 
"

 
 &2

2


   " 
  


/ 0 #



  

" 
  


2 /
0


& /
, M 
    2
 -


#
2      
$


%#
 
2   
 
.  M 
 
  2
"
 
   "
-
 
-
  











  0

 "



E

!

 

 

& M 
  2
" #   ,
& '
 M 
"  

 2  
  
 
-







  


&   


%&.2   
-   #
6 4
&/&   /
, M 
  2       
& 
,  

 M 
 
2
 "    "

½  ¾     

 



  
#-   - "

& 


 2
 "  
-#
) 
2    
+
)

 & M 
  - 
!

# 
 
 # "-







 


-
./

2



  
 "



1   




 "-

 
 "

- 






&

2






 
  
 #

-

 
 "
 # 



   

  
  - 


 
 "   E
 


 
 
6 " 


, #A% 
'&2    
 ; # 
 
   "   


- # 6

  6   
  ; #  
 
  
  
  % 0
  ; #
 

     

   

4
  ; #  /

 

  



  ##A% 
'&2

 A

  

% 

 
  

; # # 6 
 
  

 
 /
" 
0 4

  

189

Chapter 3 Glossary

&&)
'& /
  M


 




2 " 
 
"
 #
6

     
 " "
 7" "  

 





# 
 /"
 "
#
60

 '

2   
%
.2
 
  5 /  0

   

"
  # 6 ; #  
E

"   
" 
 
!

   
" 


  ; #
 
!

!

# A%2
# 6  5 /  "  0 #
 
" 

  
" 2 
( !
  " ! 

  ! #  <  !

  " ! 

 
&! #  <  



 
 A%  5 /  " " 02
 
 # "-  !
"2

!

!

)

-

" 2 
2
)



 
" 

  
!

  

" 

 A%2

# 6  5 /  " " 0
" 2 
( !


     " 2 
2
) 
 
A
/0 5 ,







)

(


¾

/

0 5 ,

#  
" 

  
!

  

" 


¾



  
2    "   




 

!


    


 
A

  
  

 





2



    #  
  
 "

 
A

 



 2


  

  M 

2

        !  
6 
 





2

 
 
   
 
    
   


 
 


#'
  /  0 M 
 
2         
 
# /  
$  0 M 
"

 
 &2   

" 

 
E  

,/
0



(  /
0

/  #
 =>  
 0

#
2
  ! 

# 
  
  # 



   /
,  M 
  2
"  
 


 
      


 & 

2

E


 !


 &&) 


2



 


 
 


"

 /



 















  ;"!
 !


0   


  

2
 

! % #2
 
 
  % 
2



 
 
/   

&02











  #
6 
  







  






#&
 M 
 
2


   
  
  



#
 
 
"
      

  @ 


190

Chapter 3

DIRECTED GRAPHS

 /

M 
 
2
" 
   
  "  " 
 


A, /
&  #2   #!  


!
#

 &

2
 
   #

 
  

! & /  02
 
   # 
  /0   /0  5 



!
'&)#2
 
  #


  " 1 
 
  

  " /!  
   %  
 ,/
0  ,/0 5 %!  


 
  "

    0

  &
M 

 2


 #   
      ! #
    
  


 '
 M 
"  

 2

  


  


 
  
   -

2    


2
  
"
  " ! 

   !  

  "  " !  
 - 
 .
 

 
        
 ! $

!
 $  

! -#
)2
    # " " 
 -  # 
E
 


 - 

 & 
2   # / 0!  
#/ 0        E
#

   ! A  

A% %
.2   &

 
"  
 
  2

   " 
   
 /!
 /0 #    0
 
0   
/

0
/ 



 B /     0 M 
 - 
2   -

/            0! #       "  ! 

!
 

  
  
 


 /
2         *

 '

M 
"


 
 &2   

"  &




  

-
   
-
E   (
/
0




M 
 
  2
"
 
   "  " !  
-
 
-
    


 '&   M 
  2 
   
 

 &

2
 
 #  
-

  
-

 &

2
 
 #  
-
  
-

 

   M  
" 
1  - 
 



/       

0


" "   "   2 

" "" 
 "  /!
 /  0 " 0 

    "   
1  " 


 


 . M 
 
2
"    < 
  

   M  "  2    ! A  H

 M 
 
2
"    < 
  '
 M 

   
2
 
 
 


 "
   



191

Chapter 3 Glossary

 .
)&2

  /0 
 #    
 #

"
   
  "!
 
 




&

  
  2



 
 
#  
  

 

"
! 
- -     "
 # 
 


  M 
 
 2 

 
   
  



2       
   


 
  M 

2
 
  
 
 
   


 
2
 
 
 
 
    

! %#2
   
  
   "

   %    # "

 
   



 &) 


2

 
  # " # "
 





! !  
 
   
   # "   


 &) 
'&


   #  

    
     
  
  
   
 

)'&  M  
 
 
 
2    ! A  
 &2  
 

& & 
    2

 

 


    
 
 #  
 


2        
 

 
,2
4
 
 6 5 /- 0  ,F
 F! # ,F   




 - ? - 5 !  

 
  /! 
1
 
 
   
0


2

  
  
 # 
 
  "   








 '&2
 
 
   

$  
! B
 

2

 
 
 
E    ! A  
!

 '&2
 
 #  "- 
  
   #  -
!

 ½
 ¾  
 " "  ½  
 " "  ¾ 

! %#'&2
 
  # " " 
6
   
 %

  "        6 ! # %

! #2
 
   ! !
 -"  



  / &
 M 



"

  
 &2  

  
  & 
 


  /

2
 
  #!  /  0
 / 0

!    /  0

  / 
  M 

2
 
  
 
 

"  


  /
&  #2

   #!  


  $ ! 
#
 #$ ! 
#$ 

  / 
2
 
  
   "     "
! !
 $ ! 
 
 !
   
 $ ! 
 
 $ 


 
M 
 
2



" 
  
 "  " 
 -

 &
&

2
 
  #!  


  "  !  


 #  


   

192

Chapter 3

DIRECTED GRAPHS

/
,# M 
 
   
2
"  #  
  


 
 
     
 

%  M 

   
2

   "    
! 

   
    
<  <

Chapter

4

CONNECTIVITY and TRAVERSABILITY
4.1

CONNECTIVITY: PROPERTIES AND STRUCTURE
  
 

     

 
  

     

4.2

EULERIAN GRAPHS
   
 

      

4.3

CHINESE POSTMAN PROBLEMS
   ! " 
 #  



4.4

DEBRUIJN GRAPHS and SEQUENCES
 $ %&'    (  '

4.5

HAMILTONIAN GRAPHS
'  ' )*   

4.6

TRAVELING SALESMAN PROBLEMS

 

4.7

  &    +' $

FURTHER TOPICS in CONNECTIVITY
  
 

     

 
  

     

GLOSSARY

4.1

CONNECTIVITY: PROPERTIES AND STRUCTURE
  
 

     

 
  

     

 



 

 
 
 
 

 
  
   


Introduction

     
   
    





 
   
! "  
  
 
 
 
 
 #     *,-* 

 
   
 $%  "
      !   
 

     Æ 
$
  
  
    &   
  '
(
 (&% 
 )!
 *
%    
 
+    
 
     
 

  


 
   
       

 
 
  
  
  !&
  
 
 
   !    !    
   

      &  
  !  ,    

  

( 
 ,  

 

   
 
 
 
 -. / 0 -*
110
 -2 340 -+ /0 -*
2
110 (  
 !
  
      #&
 
 


   5'
-6770 -8970 -$
9/0 -8* 3/0 -(/10 -(/ 0 -(/40 -:/0 -;/70 -2<//0

4.1.1 Connectivity Parameters
"   ,    
     

  
 
  

  

 
!

  2


   
  
# 
  
   = >
  = > 
  ; !   
    !   


   
 
  
Preliminaries
DEFINITIONS



'  
      # 
!
& ! 
 
  
  

 
     
   

Section 4.1

195

Connectivity: Properties and Structure



' 6 
 
! 
 
# 
 !        

 
    



' : ? =  > 


    6
  
    
  
 
      
     6
    
 
   
#      "  ?  !   !   



' : ? =  > 


 
  6 
    
  
 
           6   ?
=    >    
 
#     ?    
  !   

 
  



'     
    =
   > 
 


#    
   

 
 !  @ 


'




#

 


  
 




 
   



'       =  > 
 

   
   

 
 !  @ 





'    
  = > 
 

     




 

 
 

FACTS



' A
 

  
 

 
 !
  

  

 

'

  
  
         

Vertex- and Edge-Connectivity

6   !
 B
    

   
   


 ,-


 '
-


DEFINITIONS




' 6 
    
 = > 

       

  ! 
  

   

 




' 6   

= > 


 
      
  ! 
   
  


  +  #  
 !   C D
   
  "      
 = >

= >



= >
 = >
  


EXAMPLE

'

(   !
 #
 

 ! 





? 

? 

196

Chapter 4

Figure 4.1.1



CONNECTIVITY and TRAVERSABILITY




? 



? 

FACTS



'

?

? 1 
  
 
  

   


     

 


  

 ½

 ½




   


½

? 1 

½

 


 $
 


  # 
   




?

?



'

"

?

! 


'

"

?

    
 



"


 



 

 "   
  
 




 

   8 
  ! 







 
 


½
=

> ? 1

  
 


 !    ! 



'

6 











     
  

Relationships Among the Parameters

 

6

   




 !   ! 

Æ

 

Æ

=

> +  # 



 ="      

  
 

Æ

=

>

 >

FACTS




'

-+0 (
 


'





-*
730 (
 




! 

?



?




Æ 
?







 
 1




  # 





DEFINITIONS



'
  


 
  !
? Æ 




'



 


!  


!

'

?






   
  ! 



 


  
  








?

Æ




  






 

 

 



 
  

  
 !  
   


  


 

Some Simple Observations
6 !  

   
   ,  

FACTS




'

AB
 


  
  
 

 

 
    
     

Section 4.1



'

"





  

    




'

197

Connectivity: Properties and Structure





 

·½
   

?

! 

  

 






E 
 






       ! 




    





'

A
 & ! 
 
 
    

Internally-Disjoint Paths and Whitney’s Theorem
DEFINITIONS



'



 

  

 


# 
      
   ,



#  
 




'

6 


 

=

½ ¾    

 5   
 


 >


   







 
  

 




   ! 
     





#   6

=

>

=

> ?

 



 
?



FACTS



'

-+  0  



  




'



    
  
 !

+  %    
 
   
 
&

'



!  

 5  
 
 ! 
  5  


 


! 
 
 
  
& 
   
 !
 

  
  

Strong Connectivity in Digraphs
( 
     
   #
  #& -:/70 -*
F
730
-8
210

DEFINITIONS



'

"
 


 



'




 



  







!
&

 
 
  





  
 


 

!
&

 
  

 
 !
 







 
 





'

(
   


 


    




?



=

>

 ,
      
  ! 
 

 
  

  





?

=


  

  

 

>           =


>

  


! 
  

    





'

:


   
 6

 
 
 

 

=

>
 =

 

      

 
  
 

>  
C D

 

=

 

>   !   

198

Chapter 4

CONNECTIVITY and TRAVERSABILITY

REMARKS

'

"  #       
  
 ! 
 
 ! 


  


   

 " 
 
 = > ? = >
  
  

     
 
   !
 

= > ?
= >

  Æ 
 Æ      
 
 
 

 
  6       ,
 Æ ?  Æ   Æ 

'

F 
  
   
  Æ   6 !  
  2
 *

   

  =
  > (
 9
FACT

'

-2*
910 (
  





 Æ

  
   




 
    

   !
  

    !
? Æ 

?
? Æ

An Application to Interconnection Networks

6   !& 
 
         

  
= > 
  !  
    !  
  
   
   &
   = > 
- 
     
 
 
 
   
& 
    

   !&   
  
  8 *
  
 -8* 3/0 "   
 # 
    
 
!& 
  

    
 6    B  
  = >


     !&  Æ  
   

   B  
   = >
 
 
#  
   
 
!&
   
   

4.1.2 Characterizations
+

    !   
 
 
    
 
  
 
   
  
  5    
  5   

   "   
  
 
 
  6
  

 

 !
  
  5    
  5  
 + 
!
      &       
 

   
Menger’s Theorems
DEFINITION



' : 
   ! 
5

  
 

=>         =>  
    
! 
  
 
    @ 

  (
 
  
5

 
 
  
=> 

?   
  


  =>     

Section 4.1

199

Connectivity: Properties and Structure

  (
 !
 

  5   



  => 

 
#   

FACTS

'
'

?   => '     
5

-$%  $90 (
 
  
5

  
 
=> ? =>

'  => 
 
 
  
 
       
 
  $
 
 
 
 ,  
 

   !  B
 
(
 


  


'

 '

-$
90 A
   

 
 

5

 
!  => ?  => =>


  

  (
 
    
  
 
=>      
  ! 
  =  

> 
 
    @ 

=>   
#    '
-'.  


'
'

   
=A

  $% > -A(47((470
=> ?
=>
(
  

 



= > ?  
=>

REMARKS

'

. 

   $% 
  
 # 

 


  


'

6  

   $%  ! 
  (
 (&
 -((470   & "-/&  F!& )!     
 
Other Versions and Generalizations of Menger’s Theorem

"
      
 !  #  
 
 
 
  
$%    #
 -. /90 -(/40 -$3 0  
 



   $%  
    -;10
DEFINITIONS




'


2
    
   
  
   
    
#
   
    
#   
    

'

         =  = >  >  
  




#   

'
'

 
  
 !   ?  
      

     
     

  
#   = B
  
  
 
 
#   >

'




#  
     !  
 

5


  6 
#    = 
 > 5   
  
= >
    
   = > 

     = >

200

Chapter 4

CONNECTIVITY and TRAVERSABILITY

FACTS

'

6     
  

 



   5 

'

"




 6
 


=

> ?






=

=





   




=

>

>

'

6   $      
 B
  
   

" 
  #  #
  ! 

'


=

> ?

 
=

> 

2

-2
70 5 
 (
 4    C#
D  
 

   

 

!


=


 
=


  


    
#    
  5 


       
  





 B
   
# 

>

>

=

> 
 :
G
 -:970 5 



  
=

>

>  8 5 ! 
  $


'

-$
93$
930

   
=

>

=

> 



 
 
=

>

=

> 

REMARK

'

6 
 

   $%  =(
 3>  
  
  (
 

 
& 







   
 
5
 







 
 

Another Menger-Type Theorem

 

(
 
 
 



  
    






  


=

>   
#  

 
  5 


    
  B
 

5

 





=

 
 




 (
 
  

>       
  


 




  
 






FACTS

'

6
 #
  !  ! 
    B
 

*!
 

   
?

=

>

 =   




> ! 


    
   
=

>

=

>

=

> ?

=

>

6  $     B
 

 (
 /


'

-AH
99:F 930 6 
#    
  5  


  B
       
  = @ 

 











> 
 




Whitney’s Theorem
"
 
  # 

 !
 
  
 
  

      # 
 
 ! 
  5  
 !
!   
  =(
 >



  $%  ! 
  


&
  
    
  
  



 
 !  !


 
  +   " 




    

 
 




 


FACTS

 '

-+  %  +0   

 


 
 


 

   
  

 




=


  
   
   

 






 
  


  5 




 >






Section 4.1

201

Connectivity: Properties and Structure

   
     
   # 
 
    5 

' =A
   +  % >  

 



    
 

 



  
 =  > :  
 
           6  #    
  ½ ¾      



    ?        
  
 ½ ¾    
 
  5  = 
!  
#   >
  = >   ?     ?     

' =6 (
 :
> :













Other Characterizations

   

 
     
 !
  5
  (
&
 $
 6 5 !
 
  :
G

  2 I
 =!
!&  >
 


 (
   



 
  
  
   
 =(
  >
FACTS

!    E 
     
  

   
  ½  ¾    

  
  ½ ¾      

½ E ¾ E E  ?   

   ½  ¾       = >  
   
  ?  
   
 = >       

' -:992 930  


! 
 
       
   
    
  ? =  >        
  ½  ¾      
 
  5  
 
 = > ½  ¾      
   
       

' -/90   


4.1.3 Structural Connectivity


*     
 
 
 ,
  
   

      


Cycles Containing Prescribed Vertices

6 , 

 
   . 
 !  
  (
 4
FACTS

' -. 710 :
 



 



  
   

6



  


  
 !     6


 

 

   !   E 
  
       
 
!    ? 
  ! 
 

 
     
 

' -+
$790 :

Cycles Containing Prescribed Edges — The Lovász-Woodall Conjecture

:
G
 -:9 0
 +
 -+990  5 
 
  

 

  
 

     '' '
 =   ! 



#  > 
     
       

 
  

202

Chapter 4

CONNECTIVITY and TRAVERSABILITY

REMARK



' :
G
 -:9 :990 , !  +01-(' .
      ? 
6 5 !

 !      ? -A2 34:/10
  ? 4 -
/70
B  *I
&

 6
 -*
630


  
   
  
  = E >  =!           > $
   5   
    J
!



 
   

 =   !  
     
>
FACT

'


  
 !    
 


  
   
    

        
 
  
-J
1
J
1
J
1J
10 :

 
     6

  
      
 

  $
 
      

Paths with Prescribed Initial and Final Vertices

2

 !      
  
  
  #   
 5  
  =    >   
   

  $% 
$%    !
  
 
   
 
   
 5 
,#    
 
       =    > F! !   
#   
 !    
 
DEFINITIONS

'  
  
 
   

 
 
 
  
 B
½ ¾      ½ ¾        @
   # 
  
    ?
       
   


#  5 
'  
  
 
   

 
 
 
  
  


  =   >  # 
  
         
   








  5 

















'  
  
    
   
 ,
  
 
   
     

  5  
 ! 




' 6    

   
  
  !  


 
  

FACTS

'

'

  & 
 
!
 =  >    
   
-:
$
910 -H910 =  > ( 
   # 
   => 

    =>     &


 '

6
 -631
0
    

   
 

    & 6    ,     
   
 
 

 
        -340

  (    =>   
   
 
 => 
 
  !
&   &

Section 4.1

203

Connectivity: Properties and Structure

CONJECTURE

   = E > ? => ?  E 

-631
0 ( 
 
FACTS

' -;&3 ;&34;&390 "      ½ ¾     ½ ¾    
 = 
 
  >
  
  !     7

=  >   =    >  
# 
   
        
   

   5 
' -*/0 ( 
     = E >   E 
 =>   E 
' -;&33;&/1
0 ( 
    

= E >  
 = E >   E 
=> =>  
 = E >   E 
' -610 A
  =>  
 =,  (
 /> !   
  #


      
   &
' -/90 : 
    
 6 
    =½  ½ ½>
=¾  ¾ ¾>     =     > ! ½  ¾      ½  ¾    
  
    

 
 ½  ¾     
  @
     ? =   >  ?      
#     5    
    
   = >  ?     
=
>





























Subgraphs

*  
  

     =(
 9> 
 

  
    

   
 =(
 3> *!
    
#  
   

FACT

' -$
9
0 A
 
     
 









 

REMARK

' 
       !
 
  6
  -6330 " 

$
 
 
  

     
  


  



 


 

4.1.4 Analysis and Synthesis
   B      
 
     ! 

 
  = > 
 

 C D  
  

 
  
    
 
       
6%  !  
 !  
   
 
  ! 
&
 +
    
  
  !      

  ( 
   
     

   
   


!

  


  
  

204

Chapter 4

CONNECTIVITY and TRAVERSABILITY

Contractions and Splittings
DEFINITIONS



' 6    
        ,
      
  =&   
5
   
      ?   > :

  
     
      
   
 
 
  




' 6 
 
   
 
' 
#  !   Æ  


    
!

   
 
5
  
 !
5

 
  

5
   $
   !
    
 


  ? Æ E  ! 
&


   




' (
     
    
# 
 
 
5  

#  
    
      
 ! ½
FACTS

'

" 
  
  
   
#   
 
  
!
 

 
 
 =
 
>   " 

 ! ! 
 ?   
  Æ  

   


'

'



-6310 A
   
      

 
  

-630 A
  
-  =   >

  
 

 
 

 '

-670 A
   
 
  
 
! 
,  B
 
#   
 
  
EXAMPLE

'

" (     
 "       ! 

      
 
#   

Figure 4.1.2





        !"
 "

REMARKS

'

" 
    
  #    
  

Section 4.1

205

Connectivity: Properties and Structure


'

6
  (
 3  

   J
!& %   

   (
 3 

  
  6%  =(
 41>



'   6% 
       
    
  


 
  #
   (

 
    5
!   
  -J10 ! 

   
 
 
= !>

'

(
 41 

    !     -670'
 

   
!   

  ! 
 

 

   

 
   
   
    

'


 -9 0 

  
    
  
 
 
       
   
 
 B  (   4  
    *!
 :
G
 -:9 0
 $
 -$
93
0 

  
 
  
 =
   
!>  
  


 
 
Subgraph Contraction

6 
  

 


 
 
    
 
DEFINITION



'   
  
  
  
      

  
    
 
#  
  

FACTS

'

 /
  

 
  


'




-$;/ 0 A
   
 
  !

-6630 A
   
 !     
 

 
   

'

-J110 A
   
  
 
   

 
  

   

CONJECTURE

-$;/ 0 ( 
 
  
  Æ  
  

 
 

   
Edge Deletion
DEFINITION



'  
  
   
  
   #  
  

 
   
      =  #> 
FACTS

'

 

-$
9 0 A

!  


  
 !     
 
  E  

  =! >   

206

Chapter 4

CONNECTIVITY and TRAVERSABILITY

'

-;&330 : 
   
 !    
 :   

½        ?  = ?  > 6
=
> 6 # 
     
  
      
=> 6 # 
    
 
       
' -;&/10 : 
   
 !     
 "      
   
  !    ?    ?   = ?  >
     
!     E   
         # 
    

     
' -*;&/0 ( 
      # 
   
 
 
!
  
   
 
   
      #   !  #  




 

REMARK



' (  
   
      
  
 ;&

-;&/40
 


 B
 
       
   
 
     
     
Vertex Deletion
FACTS

'

-J
: 90 A
   
     
 





#

  
     

' -630 A
 = E >  
 

  =>   !
   
  

 ' -A390 A
 = E >  
  
 

    !
   
  

REMARK

'

(
 4/ !
 5  :
G

 6
  (
 /  
 

Minimality and Criticality

 

  B   

       



    
 
#   
 =  
> !        

 
 

#   

  !    
DEFINITIONS



'  
   
  
   

     = >   
 
     =  >   
   

   

= >     
   
=  >  

'


#  
 
 
 
        => ?    => ? 

FACTS

'

-$
9$
90 A
   
   =


 
  E 
    

  >


 

Section 4.1

207

Connectivity: Properties and Structure

'

-$
90 A
   
  
   
 


#  

'

 


'

 E 
  



A
   
  
 
  

#  
 ! 
   !  
  
-*
30 A
   
    
 

 


  

REMARKS

'

*
  -*
7/*
110 
  #  

#     
   



        
   

 !
 
  : & -: 90 8  !  
  $
 =(

7>

 



' (
 7  
  *
     


  =
  > $
%
 =(
 7>
  #  
     6 #  
 

 
#  
   
  

  J

-J
9 0
Vertex-Minimal Connectivity – Criticality

$

 
 -$
990   
  



'


 '
-

'
  
 ! 
 
 !  

 
 

DEFINITION



'  
  
   

    
 =  >    


#   !      ! 
 =   > ?    +  ?  !  
   

  

   
FACTS

'
'

-$
990 6  =  > 
    


½ 

6 C&
 
 
D =
  ¾·¾  

 
 -

 0> 
=  > 
  
=  E > 




' -330 6  
   E
     B    
 

 !   

'

-$
990 "

=  > 
    
  7 ¾ 6  


  ,  
  =  >   
 




REMARKS

'

 

 =  > 

 !   5
  

    -$
3 0

'

(
 77  
  5 


  ·½     =  > 

 $  ! 
  

  !
 ,
 
   =(
 79>

' (
 73 !
 
   $
   
 
 ,   

! 

208

Chapter 4

CONNECTIVITY and TRAVERSABILITY

Connectivity Augmentation

+         
  -(/ 0 
      

 
* "  '
-

 
*  * !
 



? =  >

 
  
  
   ,  
 
   
 ! 

 
 
 ¼ ? = 
 >    . 

 
        
 
 !& 


  

  !  
 
 

     
! +



 F
&

-+
F
390 
  ,  
  
  
   


   "  
 
 
 




Æ        

 
 
  
   
 



    6 
 B    
 !
 
  -(/0

References
-. / 0  

  .  $%  

  
 



 =// >  4K47
-8
210 H 8
 H
 2 2 
    L
 : 11
-8970  8  

F! <& /97

  

     

 

 
  
  A   F *
  

-8* 3/0 H  8 F *
   
 :
 
 
 
  !&  
  
4 =/3/> 19K
-*
730 2 


 ( *

  2
 !    
   7K
7      
  

   !""# 
   F! <&
/73
-J
: 90 2 

  J


 .  : &   



$
%
 =/9> 7K73
-:/70 2 


 : :
&
 *
 :MF! <& //7

 


 
 &   
 6  A   



-. 110  .   
    A   2

 6#  $

 
L 9   L
 F! <& 111
-. 710 2  . 
 "

& 2



I
 
N  $
'
 =/71> 7K34

2
  

-A(470  A
  (  
  A 
     
#  )!

!& ()*  
( 
  "6K =/47> 9K/
-A390 < A
!
      
 !    
=
 
 +
 
  %
,  =/39> 9K99

 >

Section 4.1

209

Connectivity: Properties and Structure

-AH
990  A  . H
&
  
 2  
  

(***  
    %  =/99> 71K 7
-A2 340  : AI
 A 2 I
      
 

 
 
    $

7 =/34> K
-((470 :  (
 .  (& $
# 
 )! 
!&
$
3 =/47> //K 1 


+


-(/10  (
&
&  
   
  K

   9K11  8 J
: :
G
 * H I
   5
 =A>  -.  /0%(0
  8  //1
-(/0  (
&   
    
 B 
   $
4 =//> K4

%($ +


-(/ 0  (
& 


    !&     K7 
H 8 
 J2 $ =A> $    1 %    
!!2 6 N
   $  
   // 
-(/40  (
& 

 !& )!  K99   2

 $
2I

 : :
G
 =A>      A
   8L
//4
-2
70 6 2

 $
#  $   I
 

  (
&

2
  $
%

 =/7> K9
-2*
910 . 2
 ( *

  
   

  8  =/91> 14K 

0
'
$


37

-2 340  2      
   
  N
   
 
//7
-2<//0 H : 2
 H <
8

 ///

 
    ( 

    

-2 930 A 2 I
  ; 
   
   
  34K /   
  F *
 
 /93
-*
110  *
  $ 
   ,  
 +
 
   4 =111>
3K4
-*
7/0  *
      

4 

+
 
  9 =/7/> 41K

-*
30 < ; *
  OB P  # G 
  
  G
+
 
  %
, 1 =/3> K1
-*
F
730 ( *

   Q F

 . 
! 
  
 5 4 .
  /73

(   3  4 

210

Chapter 4

CONNECTIVITY and TRAVERSABILITY

-*
630  *I
&

  6
      , 
$
 =/3> /K 

  

-*
2
110 $  *   

 * 2
!  
# 
 '
!     B +
   =111> K41
-*/0  *&  Æ      
   !
&   &
 
9 =//> K4

 
 

-*;&/0  *&
 * ;&

 #
 
5  $


    ,
     
  
  
3
=//> 4K43
-H910 *  H A  L
   
 Q

 I
 2

$

39 =/91> /4K1
-J
1
0 J J
!



  ;  !  5    
  '
:
G
 +
 5 +
 
  %
, 3 =11> K 
-J
1
0 J J
!



  6!      ' :
G
 +

5  
-J
10 J J
!



  A#
   !     
' :
G
 +
 5  

 
-J
10 J J
!



    :
G
 +
 5  

 
-J
9 0 6 J

 F  *
 %     
  
 +
 

  %
, 9 =/9 > K 
-J110 $ J  
  
    

%
, 31 =111> K 3

+
 
 

-J10 $ J   
  
    
 
 
#

   
  
3 =11> K1
-:
$
910 . 2 :


  $
  ;  #   
 ,
  !  


   &    
0 $
% 1 =/91>  K71
-: 90 .  : & $  
    

=/9> 93K3
-:9 0 : :
G
  4

 
$



+
)  .
$


4

=/9 > 3

-:970 : :
G
 ;  
    
 
  $

%

3 =/97> /K3
-:990 : :
G
     
   


%

1 =/99>  K4




 $



-:/10 $ L :
       



/ =//1> 4K 

 

Section 4.1

211

Connectivity: Properties and Structure

-:/0 : :
G
    

 //

 *6    A   F *


-:F 930 : :
G
 L F
 :


 $ .  $
  

     
$

/ =/93> 7/K97

$



-$
90 + $
 $  
 
 &

I
 2

,# / =/9> K3

-$
9
0 + $
 A#  
 
I
 6 
  2
 

I  J
  
$
%
7 8
  9 =/9> 37K/9
-$
90 + $
 A&
 2
    
 
 
I
 2
  
$
,#  =/9> /K 
-$
90 + $
 2
  &
 Q

    2

14 =/9> /K
-$
9 0 + $
 J 
  +    2

7 8
   =/9 > 39K1 
-$
990 + $
 A & I
 I

 K4



&   2



$
%


$



-$
93
0 + $
       
  

$
 =/93>  4K7 
I
-$
930 + $
 N
  $
# 

 &
 5&
,# 1 =/93> 4K7
I
-$
930 + $
 N
  $
# 

 & 
,#  =/93M9/> 39K 1





+

+

$



/ =/99>


  
 
$


 
$


-$
9/0 + $
 

  
  ,  
  77K/4 
   * 
 ! 
 2 3  *  
  /9/ : $

 : F  3 /9/
-$
3 0 + $
 ;    
  
  3/K/3  ! 
 

  4(  5 6789: 
   6 ;F /3 
-$
990  8 $

  H 
 ;
$
1 =/99> 44K7



  




 


-$3 0 + . $
      $% 6
=/3 > 9K /

  

+
 
   3

-$;/ 0 + . $

 J ;
 
       

 
  %
, 71 =// > 13K 
-$90 J $ Q
  J
 

+


-
$
1 =/9> /7K4

-;10 ;  ;
 L

   $% 6  : + 8 &
  H
+  =A> %    
   



212

Chapter 4

CONNECTIVITY and TRAVERSABILITY

-;/70 ;  ;
 

  
  
'  
 

'   7 =//7> K4
-;&3 0 * ;&



  
  
 +
 
  , 9
=/3 > 4K9
-;&340 * ;&



  
  
 ""  9K4  H
&


 =A> '       +   , 
   /34
-;&390 * ;&



  
  
 """'  #  
  

 
  
 =/39> 4/K3/
-;&330 * ;&


     
 +
 
  %
, 4
=/33>  4K44
-;&/1
0 * ;&

 A
    
  !
&   &  
 
 
7 =//1> 9/K34
-;&/10 * ;&

   
   
     

  4 /K44   8 &
  * =A> 
    
  
     
L
 *  //1
-;&/40 * ;&

      
   
   = E >
  
  
  
 =//4>  K91
-340 F 
  .   2
  R

   4K9 
% 8     !9: =2
! /34> : $
  : F
 1 
  N
  
  /34
-
/70 .  
 ;     ,
     $
4/ =//7> //K
4
-9 0  H 
  
 ,
  
3K/3

 
 +
 
  9 =/9 >

-330 H H    
% 5     
  
 ;6
 *  
#  =/33> 794K793
-/90 S <   
 
   $%   
 
 
    $
94 =//9> /K/7
-610  6
 6 AI G
      
  
 

      =11> K
-631
0  6
   & 
 * 
 +
 
 =/31> 9K93
-6310  6
 

 
 
   , 
 ,  
 +
 

  %
, / =/31>  K9
-630  6
 F 

       
 +
 
  
4 =/3> 4K4 

Section 4.1

Connectivity: Properties and Structure

213

-6330  6

   
 
   /9K  : + 8 &

 H +  =A> % 
   
   ((( 
   :
/33
-6630  6

 8 6 F 

      
 +

 
  ,  =/3> //K 
-670 + 6 6      
 ' 

<
(

$
 =/7> K 44
-6770 + 6 6  8    
 N
   6  : /77
-+
$790 $ A +
& 
 . $ $  
 
  
 

+
$
/ =/79> /K3
-+
F
390 6 +



  F
&

 A 
 
  
+


% %
4 =/39> /7K 
-+0 * +    

  
  
 
+
$

4 =/> 41K73
-+ /0  +  6 #   
 
   K  " $ *

 L J& =A> 0  '  

%
 7/  $

 

(
      //   8  //
-+990 .  +
    
   ,  +
 
  %
,
 =/99> 9 K93

214

4.2

Chapter 4

CONNECTIVITY and TRAVERSABILITY

EULERIAN GRAPHS
   
 

      
 8
  .,  
 

 
 
     A
 6
 A
 6 A
 
 ;   
  
   2
 2

4 L
  6   A
 6
   .  
7 6
  A
 6


Introduction
A
 
  
     JI  8  ' ( 



     
  
    (  =
>
A

C

D

B

(a)

(b)

Figure 4.2.1


  
 

    

      
!
&

 
 
     
    !
&  

  
 

T : A !

       97 -A970U 
 
 
 
 6  
 
 
!
  C   ,
D  

   
 (
 #
 
  
 

 
   
-(/1 (/0

4.2.1 Basic Definitions and Characterizations

  6    

  
   # 

 

? =  > !   
#  
      
      =
>   
 

    

  
    
 
 
    
5
 C
D    C
D  CD

Section 4.2

215

Eulerian Graphs

DEFINITIONS



'  
   

 =  
> 
 !
& 
  
 
=
>  ,
   
  
 # 
 
 
 
 

 
 

 #
 

'
'
'

 
  
   # 
 
 

 
   
 
 
   
  
  

# 
 
 

"
 


#   
    
   
 B

  
  

 
  
 
 



  
 "
   ** 
  
  

 ! 

  




 B


'  
    

 =  # 
>


  
    
    
 = 
       # 
>  

    
  





'  
    

 = 
>


      
=
 >   
 
 
    
  =   >
Some Basic Characterizations
FACTS

( 
   !  
   -69 $3  +/1 (3/ (/10

'

 
  
 1 =-A970 -* 390 -L L0>

:


 
 6 ! 
 B
'

=
>

 


=>


 
 


=>



    

'
'
'
'
'

 
  
 
    



  
 
 
  
 
    

     

 
  
 
   
   
     
 
  
 
    

        

  
 ? =  >  
 
       
 =     > 
 

   
    =-9/ (3/
(/10>

'
=
>
=>
=>

(
  


%  ! 
 B


%  

% 


  

% 

      

216

Chapter 4

CONNECTIVITY and TRAVERSABILITY

REMARKS

'

(  
 
 

 
  A =-A970> ! 
 (
 =
>  
(
 => !   
    *  -* 390 6 B
  (

=>
 (
 =>    L =-L L0>

'

8 (
   
  (
   7 
 
!






 
   
 


'
'

F 

 
  
   

6    
   # 


   

 
  # 
(
 
Characterizations Based on Partition Cuts
DEFINITIONS



' :



  
 = > 6   

 !   
  =  >       !      
    
 ?  = >     
    
 
   # 
 

 ,



'      
   
 
    

  

  # 
  
 

 !     = >




' 6  

  =  # >  =  >       
! 
   
     =  > 6    =  >    
   ! 
   
    =  >



' :  

# 

  
   # 
  6    
   Ú    
     =  > !  ? 

  "
 
  

 
       
 

Ú
 Ú  
 
FACTS

'

'

 





 '
=
>
=>
=>

   

  = >
:

 =  >  
  
   = >
 
 
     =  > ?  =  > 

 
 
  


  # 
 6 ! 
 B
'

 


 =  >  ¬  =  >   =  >¬  


 
  
 
 = >
¬

¬



    

REMARKS



' +  (
 =>
   


 (
 9=>
 =
>    


  


 =    >  
   # 
 
 
     (
 1=> =!    (
 3
 /   


  
>

Section 4.2

217

Eulerian Graphs

'       (
 1  
 
 

     

!   
 
  
 # 
  
 
  

  & "-/& 
;  ,  


  
  %  U 


   %  
 
   -((70

4.2.2 Algorithms to Construct Eulerian Tours
+   !  ! 
 

    
 
   
      
  
   = -(/10>
Algorithm 4.2.1: #"$%"& $!"' (#)*

#'
 
 !
 
 
 
 
5'
 
   





# 
 
 
   
+  
   

  
 


#    
    
  

 

#  
 
 %   
A
 
      
 %  

# 
  


     , '-<  
 =
1>  !  
 
  
   
& 
       

EXAMPLE

' 6 &       
 
 
      ! 

  
 R  CD 6 
     
  ?
&  & & &
 % ?       
 !  (   6 

 
  !  %     
 

#   
   ¼ ?
&  &     & &   # 
   
         
  ¼
  
 
      

 ½

Figure 4.2.2
REMARKS

' 6    
   * %
  
 

- 

   
     

218

Chapter 4

CONNECTIVITY and TRAVERSABILITY

'

6 
  ( %
  ! !  
     
 
    
      
  ( %
 


  -:3/ 0
Algorithm 4.2.2:

$"& $!"' ( $)*

!  ' 
   =
 
  ?    ¼
 
  
:  ?      
(  ?   '  
:  ?   = >
"  = > ? 
:  ?    =  >

#' A
 

5' A
 

A

>

  ?    =  > 
  
  
A#    ?               



The Splitting and Detachment Operations

6 

 '
* 
  
 

  
   
   


 
   
          
DEFINITIONS



' : 

 ! 
#   
 =>  
    
  
 
   
 
  6 


    
!

# 

  !  ¼  ¼
5   

   
 
 
  
  
 
   
    6 
  
 
 
 

   = (  >

Figure 4.2.3

 $! "
 "!   $ 
 

' :  

# 

 !  =>  
    
 =>  =>    =>    

          
  ! 
!
         
     ?              =>
=! 
 
   > 6 
   
  

 
 
  6 
   

      = ( 
 >

Section 4.2

219

Eulerian Graphs
e3
e3
e4

e2
e1

e4

v

e5

G

Figure 4.2.4

e2
e1

v1
v3

v2
e5

H

+"
  

' 
 



'  
  
  
   
B  


  
 
  
   
# 
  = > (
    
  
  -F
9/ F
34
 F
340
FACTS

'


+** :  

# 
   

! 
=>  
  ½  ¾ ¿ Ú 
=
> "   

# 
 
      ½ ¾  ½ ¿  

 
=> "  

#
 ½
 ¿    @ &  ½ ¿  
  



' :  

# 
  
 =   
 > !  =>  
"        ½ ¾
        ½ ¿     ½ ¾

½ ¿ 
  
 
# (
   
  "   
  ½ ¾

½ ¿
 
  

'
'

 
   
    

 


 


 
  
 
    


 
 
 

REMARKS




' 6    :
=(
 > 
 

  
   
   

         =  -(/10> " 

     
!     
  
   =

 !
 
> 
  

   
 
 (
       :
  -(110



' .,     
 (
    
  
  

!   
 ! "   
 =
 
 >  
&    

  = 5  >
   ! -'
     

               
 
 


 @ 
 
     =>
     
 



' 6    
  
 
!


 
   
 

  !  
         
 #
 !  ! 
 

  
  #
 ! 

220

Chapter 4

Algorithm 4.2.3:

CONNECTIVITY and TRAVERSABILITY

,$! $!"'

#' A
 
 !  ' 
   = >
5' A
      

  
" 
   ? 
  ?    ¼
 
  
:  ?      
(  ?   '
"  = > ? 
:  ?    = >   = >
A 
    

"   

#  
  ?    = >   = >
 
  
A
  ?   
 @ & 
  
 '?      = E >    
  
A#    ?               

REMARKS

'

  K  
 
  

  
 
  
 
'
         
    
      
 
 



' 6  @ !      
 ( %     #

   
 
  
 
   1    '
  



 B 
  ( %   !        
 
 
   
   "  
 !
       :
!  

    
  = -(/10> ;
 

 



 
   

4.2.3 Eulerian-Tour Enumeration and Other Counting Problems
6 8A6 6 

 #   
 
  
  
 
  
 
  
 "     ,   * =(
 4>
 
 
   = > 
    

 

  
 
  6 
 !
 ! #
 
 !    
  +
  )   8 5 = >
 
    
  .F B 
  B  .8 5  

   
 
DEFINITIONS



'    
 
 
 

   1

 

    

   
   !   




' : % 
 
 =%> 
5
 
 # !   
  
 
 
   
   =%> ?       6      =%> ! 
 
   ,   

  ? 
 



 ? 
 ? = > 
 U

   



Section 4.2






221

Eulerian Graphs

'

:
  ? 
½   
  

 
 !

  
     
  
    ! 
  
    
  
   B      
 

 ! 
 

 #
        B





' :      6     %  

 
   
=  > ! 

 

 U  

  ½
  (

   !
½    
  

   

  %  
 5  

#
½    
 ½  
#
¾    
 

  
 (

 #     =  >      
 # 

   

  !
     

FACTS


'


 ,   * 2

 
 %  =%> ? ½    
   ?
 =%>   J @ 
 # 6   
     % 
   
   



' "
 
  
 %    
    
  B

   
    
  

'
'

%    ?          
3) -  * -A84 6 0 : % 
 
  

  
 =%>
   
 
   6   
  
 
  ! 
 

 


=)= >  >V
  
(
 
  









' (
 
 
 !  *
 
 ' 
    = > 
 
= >  

  
   
 
,#  
    
 ,



  


 '

6 8 5 
 %  

 

  =%  >> !  

 
= >    


  
 ==> ? )=> ?  


' 6 
   !     8 5 B 





    
    8 5 
 %   B 



 
    8A6 6     8 5 B 

 

 
½
= V>



EXAMPLE



' 6 8 5 

 10>

% 
 % 
 !  ( 

4 =
 -;/

222

Chapter 4

CONNECTIVITY and TRAVERSABILITY

0000

000

1000
D 2,4 :

00

D 2,3 :

001

0100

010

1100

10

01

1101
011

11

0010

010

1010

101
110

0001

1001

100

001

100

000

0101
1011

101

0110
110

111

0011

011

1110

0111
111

1111

Figure 4.2.5

 -". !"
 % 
 % 

REMARK

'

.8 5 

  
 
    
 ?   ! 
!
  
 B 6    
 % 
  
 
 
    !       .F B  6 
 

   !

   
  !    !  
B     

'

  %  
 J
 

  


 
  
  
  ! !    


 


      

 
  
   
 
  =  -S10>

4.2.4 Applications to General Graphs
"     !  
 
   
 
   

  
U   
 
 

 
    


  
  #
   !* ! * =
> "  
!  


    
 
   B
 !  


 
  B  



 
 
   
 
Covering Walks and Double Tracings
DEFINITIONS



'  
  
 =    > 

 
 


  
   





 !
&

'  
    
 !
& 
 
 
  #
 !  
 
     
 
      
   !   

Section 4.2

223

Eulerian Graphs

 '

        
!
&

      ½      
 
  ?  =
   ?  >
 
       
 






'
'

  
   
          

 
 

6   
 
 

   ! 
 
   

= >   

 
= >     
 
    

FACTS

'

:


 !  
      $ 1 6



      
 !  

 
 
     
B 


     
  
U
   ? 


   

 
 




' A
  
 

   
  
  "
 
 

      




' -
990   
 


   
  
    

 
  =
    >



' -6390 " 

 !  
 

    

 

  B  
   
 

  
 



' -6770 -6390    
 
 !   = >  1 
 
 


  

'

-L940 :

 
    = >


 
   

   = > !    
 =  >   

     

  
     
 
 
 ;
 
     (
  =
& 
 ?  = >
REMARKS

'

6    
 
       
 =.,   />
  
 
     
! 
   ! 
  ! 
!
  
!
  
  
  ! !


'

6  
  B  (
 
 4 
  
   

      
     ,  
  
  

     
  
 
   
 
  
  

  !    
 5    
 
 
 
Maze Searching

"  #     
*1 

!

 
  ! 
 

 

# 
 
    6
 %
   5   


*1-


 * = -(/0 
 #
  >

  "          
=>    
 !


 
  
#    ,  
 
 =>     

 
 

 
  
 

224

Chapter 4

CONNECTIVITY and TRAVERSABILITY


""& $!"' (
)
*

Algorithm 4.2.4:

#'
 
 
5'
   
 


  



 

 
Ú  
   
  
   
   
     
   
 
     

 
 
   

    
 



=

" 
 

>

? 1


?

=

> ?

+  =

+  =-



>

=

 

>0

=

?

'?

'?

:

>

?

>

-

=

A#

>0



=

> 







E

?

?

=

>

'?

'?

E

Covers, Double Covers, and Packings
DEFINITIONS



'


 
 







 

+

 
 
   



'

   


+

#
 !  





 
 




+



'

 .

+

 
 
  

=.>  
  


    5    

 


 


 
!

 
 




 

   

 




   

'

+





   

+




    
  

 
    
    !  

+



CONJECTURES


 !

" 




  


 !





=.>' A
   
 

.

  ' A


  
 

  

.

# 


 !





  '

A
   
 

. 


 
      


Three Optimization Problems
DEFINITIONS



6

'

:


   !  
 !  !   

   


!



!  
  
    
& 



'







'

+

6

$%


6

 


+
=

+

 


=





!
!

>  
 

=

> ?

 ¾

=

=

> ?

'

 ,


=




>

= >



6

>


 
 & 
 =$+ >

  ,
 

>    

$ %

  
& 

!
+ +

 



+
=


&  & 


>  
# 

=$+ >   ,

Section 4.2

 '


225

Eulerian Graphs

6   &  & 
   ,
   !  
  !
&
! =>  
 
   
   =
>

FACTS

'

-(370 :



   
 6 

   .


 
   
&  +  

. 
  +

 6  
  . 
 5    

 



'

-(2340 6 N    
 
  $
#  + 
 
&  
  
   
  
  

  

  $   +    
  
  
   
  

 '

-(2340 :

  !  
 !  !     "

    N    
 
 +
    $
# 
+   
&    =+ > ? = >  = > !   = >  
     !   
 = > '?
=>





'

-(2340 (
 

    
  + 
   
$   +    
 


    N   

   =+ > ? = >

'

(
    
 !  !     

 
  N    
 
 + 
    $   + 
  
   =+ >  = > 6  
 =
> ! 
 
B
  
    ==+ > ? 
 = > ? 1    >

Nowhere-Zero Flows
DEFINITIONS







' :  '  =%>  ,  
 
 

 =
> ?
 =
>
 
   =%>
¾

¾


%

6  

  

' 

' :  ' = >  -  
 

  : % 
  
   ! 
  =%>           = >
 ,  =
 > '?  =>
6  
  '   
   
)!    
 %
'   )!     (    => ? 1  
   = >
'  ' 
  )!   
  =>    
   = >
¼

¼

CONJECTURE

) *  +,

 4 )! -64 0

   =FQ4(> A
   
 

!

FACTS

'
'

-30 A
   
 

!  7 )!

-64 0 "

 
 
=>
!   )!







 
   

 

'   
 
 

!  )! 
       

  

!   )! 
      
 

226

Chapter 4

CONNECTIVITY and TRAVERSABILITY



' 6 
   
  FQ4(
 .  

 !   

  
  

  


'

-9/0 :

  

 

   '  = >
!  ! 

 B
'
=
> 6 # 
  
 +  
  
    = >

 .

6 

   #
  =>   +
=> ( 
      = >




¼

= >  



 


 ¾

= >


# =>'

  

¼

REMARKS

'

. 
  
 
  

  



  

  K A !



!
  
       

    



          #  
 


  
   
 
  


   

 
 
-2Q/ 0
 -Q/90 
 
   )!
   




' F!  )! 
 
!
 
  
 
   


 


  
    
 
   
  



=     = >>   =>
 !   
 

 


 


4.2.5 Various Types of Eulerian Tours and Cycle Decompositions
DEFINITION



' : 
 
  

 %
 
   "  
   = >
 
 
  

 

  %       

 
        
 % 
 
FACTS

'

-J470 :

 
 ! 
#   = > ?      



 
    6  
 
       5 
   

  !  ½ = > ? ¾ = >     U
      
!  
  

 
   !    
 




'

-3
 ((/10  

 
 



    
   
   
 &  

 
   

 '

:  

 

# 
   

# 

     !   

'

 6 

-A840 : %¼ 

    !      
  
 
  % ?   =%¼ > 6  # 
% 
  
   
 
  
 

Section 4.2

227

Eulerian Graphs

'

-
(/40 : ½         = > 
   !
 
 

       ?    ½             #     


= >  /  U
 
= >  /   
 
        ! 
 / 
 
   

REMARKS


'
 '

(
 3 
  
       :
=(
 >

(
 /  
  

 
   
  # 
 


  
 = -Q/90>



' % 
  
 
    -(+3/ (/10 *!
  


 
 
 
 
   
 %¼  !   
=   =%¼ >>

  
  +   (
     
   
 

  
    
 =  8A6 6 -(
 30>
Incidence-Partition and Transition Systems
DEFINITIONS



' ( 

#  

    => ?

          6  = > ?
    





Ú =>    
 ==>>  

   
¾



'       
 
 
   0 = > 
  

     0 = > ?
 =>  
  
   = > => ?   






 => A
  =>  

  
'  
  

     + 
    
    
 0
 0  
  


 !
 A
 
     
 
     0 

  
         
  


     
        0 

  

 
  ! + 
  
  
    0


    0
  

  
 

    +  
 

' :  = > 
   
     

   
  
  = > 
=  
  = >>   
     
 

 =>   = >  = > 
     
 ,

 
 '       +

 
  
  
 0  0 ? 
   
  

¾

  
 6  C
D 
    



 
       

 
    
  C
 D



'    
      = > 
 ,      


#        =  > 
 ,  =  >  =>    =  > 

   =>   = >

228

Chapter 4

CONNECTIVITY and TRAVERSABILITY

FACTS

' -J730   
 


      = > 
  
  Ú    = >

=



 
  
 


> 
 ,         

+   
 

>  # 
 
  
  + 

' -(310 2

    
 


 =

' -(310 :
 


 =

!  =>

$




 
    
 
  " =>  1 
>   # 
     
   

' -((/10 :

=

> 6 

    




 
  
 !    
    
> 
      
    = > 
 ,

=

' -(310 : 


 
 

   
 
    "
=> $   
   = >  

     
   
EXAMPLE

' 6  
   (  7 !  
     0 =  > ?   E
¼  = E >¼ '     4   7 ?  
  0 =  > 
     
!  ! 
 (
 7 
  
  
 
  

 

1

5

3’
5’

1’
2’

2

4’

4

3

Figure 4.2.6

 
/!  0 = >"!
$ $ '

REMARKS

' 6 
     + 
 

 
  

 !  =>  1   
   = >  


 

 H     
 
 - 3/0'     

 W%
 W%
  
        
 
 !    
 
 

' (
 4 !  (
 3     
 
 

 
  =(
 >
' (


 
 


 
  (
  + 
  

 

 
 =

 
     # 
     
 
 > 
   &!
 '= * .
  " 
 
   
    . 
 5
  F! Q
4 (! 5     -(:3  (33 (1 (10

Section 4.2

229

Eulerian Graphs



' (
 
 7 ! 
  #   
  
   

     
    #        
  
 
   ' (
  
       

   
    
  !
  (
 7          

&



' +  (
 7
 9 
  
  

 
    

 # 
!
  
 
  
 = -Q/90>



'  = > 
 
    
 
     -(/10 F


    


 
 
   !
     

      .,   
   
  #    = > 


 
Orderings of the Incidence Set, Non-Intersecting Tours, and A-Trails
DEFINITIONS



' 2





# 
,# B ½       
Ú    
     Ú  

 
     Ú
   1 => "
    
   1 =>  
   &!    
        



' :

 
 

 

# !  => 
 ! 
 

       Ú 
  1 => ?       
Ú  
    
0 = >          1 =>  
         ! 
      2    
     
   
    0 = > 6
 

    0 = >       0 =

>

'

: 
 
 
 ! 

  
   1 =>  
    

     0 = >      0 = >     !   
1 =>  
   !  =>    
  

    
+
         
     0
 0  
 
   

'

: 
 
 
 ! 

  
   1 =>  
   
 
   
  
     0    ?  E    ?    =
=>>



'   
   

 ! 
     
  



#

   
   #  


'

 
 =  >  


  
      

EXAMPLE

'

  
    
 
   B          !
 (  9 !       
    
   
 7 1   3    / 9 4 

230

Chapter 4

CONNECTIVITY and TRAVERSABILITY

2

1
11

10
9
8
7

12

6
4

5
3

Figure 4.2.7 

"
$   
"

FACTS

' 2

 
 


  # 



1 =>  
   =


' "
 
 
 !  =>   

  
 
  

 B


 =

>
  

>    

 ' -(/40 6     !

   

   

 
 

  
  - 
' -(/30   

 
 
 
  
U  
  
   
  
' :


  
 
 
 

  
  


 
#   Æ = >  3 ! 
   3

# !    # 

5
 


# 6 

  

REMARKS

' (
  
 3 
  
        :
=(

> B 
     
 
 

 
 
    :
 
   , 
    = > 
 
  
   
 

' 6 
   ! 
     0 = >   
 


       = >   
' 
   
! 
 
 
      
 

 U   

     
 : &!  
   
 ,
  0 = > =

 0 = >>
    

   5         0 = >

Section 4.2

231

Eulerian Graphs

4.2.6 Transforming Eulerian Tours
The Kappa Transformations
6 &



    
   
        


   


6   
     
 
 


   
  
 (

      -(/10

DEFINITIONS



'

6


     







         
           

?
?

  

  


                    
 
         
          


+ 
  
 
   


     +       
     +       
'

:

?

 



 
  



?


 B 



 


'

 



=

 
 
 


=

?

=



+ 

'

:



?



=

> ?

   



'




¼ 
=


 
  



>        

+ 

=C   !

D> 
  

?





 


 
 

  


     
    






=

>

          



> 





      

!  

     

 


=

>   


=

>  
#




 








  



 






       

    


 :

 



=  
 
  

            
  ?
  
  =    


   
 
> 6 ! 
! 



+ 

>

2


    



6   
  

 

     +   

 6   
   



>   
    

  
 

> 6  

6  


  

?

       
'

                 
                   

 =

> ?

> ?

 =

  A  
¼¼=   >

   
 
  
  













¼
:  
 
  


  
    
  
£
¼
£
¼

      ? = >   # 

 = > ?
¼
¼¼
¼
£
¼
¼¼
+   +    
  ? +   +  6
   ? = > ? = = >>
'



'













:

 ! 
  



'

6! 
 


 


0½  0¾



  

?




    
  @  

=

>   

 6

 
?

=



 
 

> 

 £ 
?

=



>


    @     
?



232

Chapter 4

CONNECTIVITY and TRAVERSABILITY

REMARK

 ' 6
  

  ,

 
 
  
  
 ! 


    
 
 
  


#
 


    * 

   

  

FACTS

' : ½
   !  @ 
  
 
 



 >  
  
 
= -J310 -&30 -(/10>




B 

= # 



 

' : 
 
 
 ! 

      = >
  
 
 
  @  = > 
 
  6  
  
  

B  

   
!


 
 

 

   + !  +  ? 
      B
  = > 


 
 
 ! 

  
   1 =>  
 
>
  
    @    
    6 

  
   
B  

   
!




 

 
    + !  +  ? 
      B

  

' :

=

' : 
   !  @ 
  
 

 
   
B  

 

 6   



 
 
  
 =! 
    
 
   
>
     !  
   
 =
  >
# 
 
 6 

     
  
 
    
   
   
 
  

' "

EXAMPLES

' 6   
  
   !  
   ?       3 =! 

  B>  !  (  3=
> 6 
  
 
 
 
&
!   
 6   
   
   ¼ ?     3 9 7 4



  = 

>
  = (  3=>>

Figure 4.2.8

' 6    A#
 4 
   
  8




  
     
    + ?     !   ?    
  ? 4 7 9 3 =! 
  B> =(  /=
>> 
 


Section 4.2

 

/=>>

233

Eulerian Graphs





¼¼

?   9 3 4 7 


    
  =( 

Figure 4.2.9
Splicing the Trails in a Trail Decomposition

+     ! 
 
      6& 
 
 ! 
 
   
 

   
  
  =  

 >       
 
Algorithm 4.2.5: 0"& $!"' (*

#' 
 

5' 
 







       

 
  


   
:
? ½              
+    
    !   ?   
  = >   = > ? 
:     
¼¼
:   ? =   > 

 
 
'?
       
 '?  

References
-J310 H 


  J  6

   A  
  
$
 =/31> 74K7/
-2Q/ 0 8 
 : 2 
  O Q
 2
 !      

     
$
%
)  =// > K4 
-(/40 : . 
 * (  6 F   ,   
 

 

  ,  
     
    


$


=//4> 1K 
-(/30 : .  * ( 
    
 


     
  
 
 
    


$
 =//3>
//K

234

Chapter 4

CONNECTIVITY and TRAVERSABILITY

-
(/40 $  

 * (   
 
 
   ,  

  +
 
   
=//4> 9K 
-;/0 2 


 ;  ;
 

       
  
"
    
   $
 $2
! *  //
-A840 6
 
 A
 F 2  8 5   
    
 
 
 %  %8   =/4> 1K9
-A970 : A   
 
 
      
   
   =97> 9  3K 1 ? ;

 " L 9
K1
-(330 (  .#       
  + 
  33 49K7

  

-(310 * (  A :    J I
&  
 .
I
   J

  +

  -
)  =/31>  4K79
-(3 0 * (        
  
  
  .
     
   H  8
 N   $ =A> 
 
 /3  K 7
-(370 * (       
 5  

 
 +
 

  - )  =/37> /K1
-(330 * (    !
   
  
 
  
  '
 =/33> 3K 3



-(3/0 * (  A
   =

 >  

 
   

 
 -  $         
 % 
/39    


$
 =/3/> F K 4K/
-(/10 * (  *   
  ) 
  
$
  F *
 
 //1

 /


-(/0 * (  *   
  ) 
  
$
  F *
 
 //

 /
=  . 

  . 

 8    
1  *        )
)57('1   %   

   $ . =A> J! 
 

-(110 * ( 

  111 /K39

-(10 * (  = > 6 
   
 
  
  =

 
 >    $
 =11> K 
-(10 * (  8 
   
 
 
  % 
    5
   $
 =11> 99K3
-((/10 * ( 
  (
& ;        

 


 +

  - =//1>  4K4

Section 4.2

Eulerian Graphs

235

-(2340 * ( 
 $ 2
 ;     !    
  
 

 
       =/34> 7K79
-(+3/0 * ( 
 A +
$
, #  =/3/> 44K71

% 
  
 
   
  


-((70 :  (
 .  (& -.  '.    N
 
   FH /7
I
-* 390  *  N
  $I
 &    :    +  
 N  
 $  12 =39> 1K
-J470  J  A  
    

 
  
  
?
! 
  B
  =  
&> $
->


%8

 =/47>
F  K7
?
-J730  J  $
 !    
   

 $
->


 =/73> F  97K31
G :
 )4 4  $4  (/ 2
 KL 
  ,
 
-:3/ 0 $ A
3/ 
-$3 0 6  $J 

   
' " 
   ! 
     
$
 =/3 > 9K 
-F
9/0   H  F
 + 
    
  
   
   
    H +  =A>   ; N
  U $  J 
/93 )
'  $
 
 
 (
  =/9/> 39K/9
-F
34
0   H  F
 + 
 .
  

 
  A 

 % 8     "  =A> /34 $
%
0  '
%
  
  N
   : =/34> 9K4
-F
340   H  F
 + 
  
  

 
 
A 
 +
0 $
%
 =/34> F  9K/
- 3/0 H  .  6   I
 2
  $
 =3/> /K1
-
990 2 
   6
  
 !  
&
&   $  5
  
)  @@/    * 
 =A> (     ;  N

*   K /97
-9/0  .          
    ) 
 H 
8
 N   $ =A> 
   F! <& /9/  K47
-3
0  .   A
     

 
 +
 
  - =/3>
9K3
-30  .   F!  7 )! +

  - =/3> 1K4
-9/0 * 
&  
     

 
  
 
 =/9/>
19K13

236

Chapter 4

CONNECTIVITY and TRAVERSABILITY

-&30 . J &  A
 

  

   
   

-  %  
   
 J $ J
 *  <
 =A>  

$
/3 : F  $
     8  F! <& /3  3K4
-6
3/40 2 6
  : P  
   '8

$
 =3/4> 39K/1
-6340  6
 
    
   
      

=/34> 7K73
-690  6 
   
A 
 +
-   (

 =/9>  K 4
-6770 . H 6  ; 
  
 
$
$  =/77> /9K //
-6970  6&  !
 
    A     
$

$  =/97> 73K7 1
-64 0 + 6 6         
   
 
+

$
 =/4 > 31K/
-6 0 + 6 6
   8   ;  
 
 
!&  
 
$
$  =/ > K9
-L0 ; L 
 
   
 B
  

     
 $
34 =/M> 37K/ 
-L0 ; L 
     
$
%



 "" =/>
K/
-L940  . L

 . 
 A     
$
 =/94>
K 
-+/10 .  +
    $J% 
  
  

 
    
$
 =//1> 9K1
-S10 H S 
  %       (   '.  F!&
6
  
   J! 
    . 11
-Q/90  O Q
 ( -.   8    
 $
 .&& "
F! <& //9

Section 4.3

4.3

237

Chinese Postman Problems

CHINESE POSTMAN PROBLEMS


  ! "


 #  



 6 8
  
 " L

 
 N  
 
 .  
 
 $ # 
 


Introduction
6  !* ! * = >      
  

   
  "


  


 
 
 # 
    ! 

 
 
  6 
  

 

  ,    

 
 2
 = J!
 $ J>  /7 -270U

  


       C D  H
& A
=-A74
0>

4.3.1 The Basic Problem and Its Variations
DEFINITIONS



'     





 !
& 
  
  

'

2

,  

? =  > !   ! 
  '
  &  & 
 &
   !  
 





 

.



  
 6 
         

   




' 6   
  

   




' 6   
  
 


&& - &&
 
  
 

&& ! &&
 
  
 




' 6  
   && $ &&
 
  
 

*,'
 
   
  
  
  
FACTS

'
'

N
 .
  
 
 =


>
$  F 
 =
 >




238

Chapter 4

CONNECTIVITY and TRAVERSABILITY

The Eulerian Case
DEFINITIONS



'




   

 =  
> 
 !
& 
  
 
,
   
  
 # 
 
 !
& 
 

=
> 


 
 

 #
 




'

 
  
   # 
 
 

 
   


'

  




 
  

 
  !
& 




'

"

 

? =

 



 



 
















 

  


 B


  
 " 

# 
 




  
 !  
 

  

> 
 


#


 

   

    

( 3   3 ( 

  
&  
 =

   

*!
   # C    
D 

>

=

>




( 3 



FACTS





'

  


 
 
   

# 

'

    


'

"  
 


 
 
  


 
 

   

=      #>  
   


   
 
 

REMARK



'

 

 
   
  # 
  
 
     $ #


 

Variations of CPP
DEFINITIONS



'

    '

6 
  B   

 
   


    
 =  
>



'

     . '

  ,   

; '

  


        
 
  



'

  .    
 '

  ,   B   

   
   
   
 =  
  
 

 >



'

    
'

"
  N 
      

    



  6     !
 
 

 

 
   
  
  #   
 CD  

 
    =   !    
  
    
 

  >

" 
!  !    @    !    


  
    



'

 
   
'

&' *  *

6 


 
  
 
  

  ! 
 
 B   

  
  

 
  

Section 4.3

239

Chinese Postman Problems


  
    6 
  
  
  

 
 

 
   
   ! 
  
 

  


  !  

 ! 


  
  
   

    
 
    
 => 
 #    
!



    
'

'

6    
 
    # 


FACTS



'

6  
  
  
 
 "







 

  , =  >
  
 
  B   


 ,

# 
 




!  

   

4

 !

4







 ! =
 ,
> 

   ! 

 Æ  
 
   
 
   
 


 ,
  ! 
  
 
 

  ! 
 
 

 


"  




U 


*   >



'







  !   ,
 
  
 

       





   &"

=



(  
  

 

 !
 ,    B 



 
   
   
  
 
 =



'

-AH90>

-+ /0 6 !  
   F 

 
   


   

 





'

6 
 
   F 
   

  
 
 


M
 
  
 !  = -2
H9/0>

REMARK



'

"  
   
     #      #



  


        

  
  

! 
    

     

   


 
 
  

  
  
 
   
 


  

  
 
 
      
 #

 
 
-A 2:
/4
0
 -A 2:
/4 0 A#
 

     
 
 

  

    -(/0

4.3.2 Undirected Postman Problems
6     2
  
 !
  
   



#     =    
        
>

6 

)
! !
     A =-A74
0> !  
 

 
  

DEFINITIONS



'



  $ 




# =$
 
   



'

 
  


 

  


    !  !  




>

 

# 

       

240

Chapter 4

CONNECTIVITY and TRAVERSABILITY

Algorithm 4.3.1: ,$  5677
#'  

5' $   ! 

!   !   

 


   

:     
  !     
( 
 
 
  ( 3 
( 
 
   ! (
 3
(
 
  
#   !   !  
 
  
 
( 
   !   
  4  
( 
   4
. 
       
     
:     


 
   


   6  
 
       


!

 
  
   !   
  



 
 
 
    
 @ !    
 !& 
A =
 -A74 0-A74 0> 6 
     

  


,   
 
  6 !  
  ( =
 -J
790> 
 
 

 ' 2

     !
&   # 
 
   

 
 !     
          =( %
 

 
>
EXAMPLE

'       (  U ! 
  ,     
6    
      
  
  

 !   

 
 !'
L#


  
  
  
  
  
  




      
         
      
   
      
   

:

4





Figure 4.3.1 $
  $!"' 

  
 
     
   !      (  
   

  
 
 !  4 6     


Section 4.3

241

Chinese Postman Problems

½  ½    
    

6 
    

  
 
    
  !      (   6 
  


 
  = 
 !  1>  
   !
& !'

                               
REMARKS

' 6  
  
   !   

 
 


 
#   !  

 !   !
&  
=    

&
   
 > !  

 

      


  = -2
H9/0>

' "
    ( 

 
 B  
  


    B     
 
  
  




 
#    
  !
   A
 H =-AH90>

' 6

 

    
 
   #  

  

 
 
 
  
    ! 

&  
  

  N       

 ! 
 
 
 
*   
  ,
   

      
  ! 

  
       ! ,
'  

 
     

 ! 
 


   
 !  
  
 !
&


   "  
    
  ! 


  
 
  
  8   !  


 
    !    
       
 

 
 !   
 =
>  
  
 !
 !  
 ! 

 

4.3.3 Directed Postman Problems
6 
  
  . 

        

 "   
      
   ! 
  
  
    
 6      

     



 
FACTS

 '   
   ! 

&

   

' 6  


 

 
      !
   U 
  
  


   6  
        
  

 

 & " /& 
; =
 >

242

Chapter 4

Algorithm 4.3.2:

CONNECTIVITY and TRAVERSABILITY

,$  677

#'    
 ! 
 !   

 


5' $   !  
   
"
A

   

 
  



( 
    ?  5= >  )&5= >

  !   
  '



   




¾

   





( 

 (

¾









(  1

¾

( ?    



( 
   
 (   
 =   > 

      
 

 
   



Producing an Eulerian Tour in a Symmetric (Multi)Digraph

  

 B   

 
 
      
  
 = 
    >  
     
 6  
 
  
 
   ! =
 -A840>
DEFINITION



'    

    
 !    

#

  
Algorithm 4.3.3:

7"!
" 
 $"
 !"


#' A
  
 
5' A
   



# 
    £ 
(
   
 

 
  
 £ 
( 

#     ? £ 
:
  
   
 !  
 
5      
  
 = > 
  

 
    
:
  
  £
 
  

 

# £  

 
   
!
  
 
 
 !   
 



    
 !
 
     


 ? £ 
  

 ' 
 
 
 

 


#   = >   = > 

#   U 
    


  
 ! 

   


Section 4.3

Chinese Postman Problems

243

EXAMPLE

'    
  
      (    , # 



#   

    
  #    
   ,
      !  ' ( ? ( ? U ( ? U
 ( ? 1
!     



     
 !  
     ,     
   =
 
  >
#



    
 !
    (   6


 
   
  !  

Æ#  
     


  ! 
# 
    
  
 
   ,
=
  >   ! 
# B'
                        

Figure 4.3.2 $
  $!"' 
 
REMARKS

' 6 !& )! 
         A
 H
=-AH90>

' 6

 

 
      
B  


 
 
 

  

' "
      B  

  
   
! 
# 
    
  
  

 
# ( 

  
 
#    
  
       (  
  
 ! 

  
  

4.3.4 Mixed Postman Problems
FACTS

' 6  # 
  $   -  
U     
 6"("8":"6< =
 -
970>

244

Chapter 4

CONNECTIVITY and TRAVERSABILITY



' $ 
 - 
 
  

 
 !  
# =
> 
#  
 ! 
  !   
 = -2
H9/0

Deciding if a Mixed Graph Is Eulerian
DEFINITIONS




' 6 
  

#
       

 '
'

 
 # 


  
  


  # 
  
   
   
  
   


# 
 # 
      
 
 B


  
   # 
  
     
  
 
  

 



'   # 
 
 ,  
     
 +   = > 
 @ !   
  +   = >  +
   
 
 = >  +  +   
 
       5  
   +
  = >  + =
 -((70>
FACTS

'

 = >   # 


 ,  

   

'

 
 
  

 



$ # 
 

 

   
 



EXAMPLE

'


   
      
 
 # 
   

!   
 

#   6 
  (   
  

   ,  
# B ½            

Figure 4.3.3

 $"
 89 +"



   6 =

> B     # 
 
  




 
 
 ,      
 

#  ! 
  
    6
  ! &         
!


   

 
 ,
  6 
 
 !& )! 
  

!        
    
    

Section 4.3

245

Chinese Postman Problems

! 
89 +"
 2 $"


Algorithm 4.3.4:

#'
 

    # 
 
5'
  
    
       
 

 
  
    
    

( 
    ?  5= >  )&5= >

 
     

     

:       ! 
 


  !  !& )!  '
   








   ¾

   

(

( 

1 (




    

( ?    

   =   > 

"  


     =   
>
( 
       
" ( ? 
;        
         
A  ( ? 
;        
         
A
:
       
A = 
 
    >
 
 
  
 
    
 

EXAMPLE

'

6
 
        = #> 
  (    

  (    "   
   , 
 
 !  #
 

# 6  

 (
 (    ,  
  
   
    =  >
 =  >
  
   ! 
 
 , 6 #   
  
 !  
     

Figure 4.3.4

$
  $!"'    !"
  !" 

246

Chapter 4

CONNECTIVITY and TRAVERSABILITY

REMARKS

 '

6  
   

     .,   
& 
  

 
 

&    =
 @ ! 
 
 

 
    >   
     
    
 5  
        

'

"   
        

 ! 
 
  
 
        
   

   
  
  
 =  > 

    
 

 !      
         
 =!   
 @>
 
 
 
   # 


   
 
 
        !  

       
 
  =
 -AH90>

The Postman Problem for Mixed Graphs

  $  F 
  
  
 ! + 

  &  

 
 
      
    ! !    
   
  
    ,* 
 *  
  
 



  
    
 !    

   ! ,
 

    
 ;   
  


     
 
     #
  *!
 


  



 !  B  @ 
  #
   ! 
 =
 -A 2:
/4
0
-A 2:
/4 0>
DEFINITION



' 
  
    
  

   



     ,
 

     

REMARK



' "  
  $ 
 
 
  
     ! 

          
 

# 
  
 "
  
  
 = (
 
 4>
 ! 
 
  
;!     

 ! 
      
  
  
  
   
M

FACT

'

  # 





  
  

  


   6    
   # 
   

       
  
      
 
 = 
>
    

 
   6 
 

     
  ,
  
#   
!  

  
 
 
  ( & =-(9/0>

Section 4.3

247

Chinese Postman Problems

EXAMPLE

' 6 
  
 

  B    # 
 
    # 
  (  4 

U
 

  
 ! 
 "  
   
   
  #      
 ! 

   F!  
   !
 
  
 !  
   
,U !
       
    
   B 
;   
   
  
 
 
  !
  
  
    ' 
U 
    
   N
     

     
 
 !
    
    
 

Figure 4.3.5 2"
 : ''"
 /!"


   "     
 ! 
!    
 
 
 !

  
    
  !& )! 
 U 


    -AH90 "
    
       


     
  # 
 
  
    @
 
 
!   
  !  
 
  
 
M 
 
Approximation Algorithm ES

6 ! 
# 
 
   
 
 
  
  

! 
  
 5     M
  
 

6 

 !
 

 =
 -AH90
 -(9/0>
    

   4      
 C  M
 
  D
Algorithm 4.3.5: "9'
 $!"' ,

#'    # 



 

5'    
 

!  M
 !  

  
  
    N     

  # 
 
  
   
  ,  
:     

;
   
    M
 
   
:     


 
   

  
 6
# 
   
    4   
 
  
   
   /#

248

Chapter 4

CONNECTIVITY and TRAVERSABILITY

REMARK

'    
 


  
          
   = 
 >  &    A   " 
   !

! 
( &
FACT

' -(9/0 6 
   
 

       A

  

 
 #  "
  
   





   A#
 7 !
EXAMPLE

'     # 
  (  7 

 !  ! 
  ,
  
 6 
  
  
    A  
   
 =      
 >     
  
 
;
     
     !  
  
    
    
  
 
  !  E 6 6 
  
   
    
 !
 *
       
 

 
 
   
  
   !  !  6    

   , ! 
   
  
  

 

    " !    E 16

Figure 4.3.6 $
  "9'
 $!"' ,

Approximate Algorithm SE

 





# 
    
  
    
A    !    


 =#/> =
 -(9/0>


   6   
         

7   
 =
 -(9/0>
    
   
   
   M
 
      4

Section 4.3

249

Chinese Postman Problems

Algorithm 4.3.6: "9'
 $!"' ,

#'    # 



 

5'    
 

!  M
 !  


     # 
 
     # 
 
:    
        
  
  
    N   
:     
     = >
  

 
   

EXAMPLE

'    
  (  9 

  
 
     A
   
  
  *!
    
    


Figure 4.3.7 $
  "9'
 $!"' , 
Some Performance Bounds
FACT

' -(9/0 6 
     
     A 
  



 ! 
 
 #  A#
 9 
 
 
  
 



 !
EXAMPLE

'         (  3   
  
 

 
 
   A 

! 
 
    A !  


   
  
! 
 
    A

Figure 4.3.8 $!"' ,
 ,  
 "& :"
 


250

Chapter 4

CONNECTIVITY and TRAVERSABILITY

REMARKS

' "  ! 
  
# 
     4 =A>

   7 =A>   
 *!
 A#
 3 


  '  

# 
  
  
! 
 
 
     
   A 
 =  
  >  ! 
 

    A !   
 ! 
    ! 
 

   A    
 

' 6   A#
 3 

 
 B  !   
 
 !' +
    A
 A 
    
 ! 
 

  @ 
  
T "     
   
 
    
  
C D 
  ! 
 
 
  


  
    +  
 
 
  
 !&   

  
 @ B   

  
@ 
  
  

  
      
 
      
 
  

 
 6 ,      
  
( & =-(9/0> !  !
 ! 

   ! 
   

     ! 
 
 
  
 ! 


  

 

   #   
' "  
 
 =-(9/0> ( &
 


  

  

 
 6    
 !
 !    

' + 




  -(9/0


  

 
 
 
 
 
   !   U   !

   
 !  ( 
/ ! 8
///  !   
 ,
    
 (
&
# 
  !
   




 L

 =
L//0> ! 

 
      6 
  (  / 
   

Figure 4.3.9  :" 
 " '   $!"' ,
 , 

' (!  

   

    $    &    
 
   *!

 ! 
  
 F 
   
 
    #    
    
 
 !   

   B & 
 6   
  
  $ 


   "  

 ,   
  
   
 '


      
 $      

  

 
  B     
   @ =
 -8
6/0 -8
6/0

1 >
6  
 
 
 
     - 
 
  
- 
  &'-


Section 4.3

Chinese Postman Problems

251

References
-889 0 A : 8

 : . 8  F!&
 L     $  

+
   '.  =/9 > 74K/ 
-8
6/0  8 8  
&
  6
  .    .   

 2
 
 %($ +      $  =//> 
3K41
-8
6/0  8 8  
&
  6
  
  2
   : 

6      
 
 .      

  2
 (
      9 =//>  444K43
-8310  8& 6   
   $ # F!&   
  (   < 
  
   
  
 % 
0  '  
 %  AA   L
 F! <& =/31> 
4 K77
-8
$3 0   , A 8
 L 
  

 A $

 ; 
 $   $ # 
   % $  
5
  >  0  '     (   %  :!   F!
<& =/3 >
-A74
0 H A 6   
  5
   )   
 =/74>  9
-A74 0 H A$
#  $
 

  !  1 L  +
)
  '  ,   %  7/8 =/74>  4K1
-A74 0 H A
 6
 (!   +
$  9 =/74>
 /K 79
-AH90 H A
 A H $
  A 6
    

$     4 =/9>  33K 
-A840 6
 
 A
 F 2  8    
 6  ; 
: 
 2
 %  %8  3 =/4>  1K9
-A 2:
/4
0  A  $ 2

 2 :
    
 "'
6   
  5
   )   =//4>  K 
-A 2:
/4 0  A  $ 2

 2 :
    
 ""'
6 
 
  5
   )   =//4>  //K  
-(/0 * (  A
 2

 
 6 
  L  
  
$ 41 F *
 
 =//>
-((70 :  (
 .  (& -.  '.    N
 
   FH =/7>

252

Chapter 4

CONNECTIVITY and TRAVERSABILITY

-(9/0 2 ( & # 
      
  +

$ 7 =/9/>  43K44 
-2
H9/0 $ 2

 . H 
   (   1     
   ' 
 +* (

  F! <& =/9/>
-270 $ 2
 2
  
  N  A

 ;     $
   =/7>  9K99
-23
0 $ 2
  
     
  +
$
)

*6

 =/3 >  K/ =   >
-23 0 $ 2
 ;  +  
     

  $  /
=/3 >  K 7
-J
J9/0  J


 2 J 6 $ # 
     

 
$   =/9/>  3/K1
-J
790  J

  
       -    
 
 F! <& =/79>
-: Q330 < : 
 < Q
  F!     .    

 
   5
   )  4 =/33>  499K43 
-$ 9/0 A $ &
 6   
   $ # F!& $
%  4 =/9/>  7 K7 3
-F /70 < F
 H  
  ; 
     $ #  

  '.  9 =//7>  /4K13
-;9 0   ;@  (

   L    '. 
 4K7 

=/9 >

-
970  *

    ;  #   A 6
  +
$  =/97>
 4 K44 
- / 0 + : 
 
 
   &    
   $ #
F!& 5
   )  0  7 =// >   K 
- :/40 + : 

  $ :        
  
$ # F!& 
   5
   )   =//4>  9/K 3/
-
L//0 8 




 H L

   ¾¿ # 
     
$ # 
  %($ +
   $   =///>  4K 
-
/0 6 J 
 ;  $ #   
  5
   ) 
0   =//>  K9
-+ 3/0 Q +  ;  +  
   A
 2
 $ 
  
=/3/>  /9K

Section 4.4

4.4

253

DeBruijn Graphs and Sequences

DEBRUIJN GRAPHS AND SEQUENCES
 $ %&'    (  '
  .8 5 2
 8
 
  2
  8 5 B
  
 F
   2   
 


Introduction
F 8 5 
    , 
     
   




= >   
  
   
    (
     

   
    
   
! 

8 5 
 "     ! 
  
   8 5 

!   
 8 5 B

 
    
  

 
   

4.4.1 DeBruijn Graph Basics
DeBruijn Sequences
DEFINITIONS

'

         
 
    

  
       
5
   ,

 
    
 & 

 #
 



?   ! 

6! 8 5 B
      C
 BD   
  

   
   
 



' "
  7   / $           &     
    &¼ 
  
      & 6      
 !  
¼
7  
    
        &   ,      7




' "
  7   / $      B     
 
  !    
           
 !   7 
 /     





'         
   
 
    
?
½
¾
¿
 

      ?
·½    ?          6    

 
 

'
'

   
½
¾   

   
½
¾   





½ ¾    



  

½ ¾    


   



?
½ 


?

½



254

Chapter 4

CONNECTIVITY and TRAVERSABILITY

FACTS

'

 
 !     
8 5 B      
  
   @      
   
   
B 

     



' 6   
     
  
8 5    &

& 
8 5 B   
 

EXAMPLES

'

1111 
8 5 B    "   
111 11 11 1 1  1 11

'

11111111 
8 5 B   

DeBruijn Graphs

  


      
8 5 B   

  ! 

 
    
B
DEFINITIONS

'

           => 
  
 !  

  
 
 ! 
 B      L#
 5  
#  

       

    
  
    
8 5  
  
  

  =>   



   
   
      
    A

  
   ,    

#
 !     
 !   
  
#
 !    




' 6 
    
8 5 
    
  
  
   





' 6     
8 5 
    
  
  
8 5
EXAMPLE

'

(    ! 
  8 5 
   

FACTS



' 6    
 
   
 
      
 



         
   8 5
 
 
 




' A

# 
8 5 
 
    6 ,     
 
  
   !      1
  ,     
     

'
'

A

# 
8 5 
 
   
A
 8 5 
   

Section 4.4

255

DeBruijn Graphs and Sequences
0000

000
0001

1000
1001

001
0010

100
0100

010
0011

0101

1100

1010
101

1011
011

1101

0110

0111

110
1110

111

1111

Figure 4.4.1  -". !"
  "" 

' A
 8 5 
  
 


' 6 
 
 = >      8 5 
 =>
    
 !   8 5 B    6   
 
     B  ,    

#  

 
 

' -".& "' -8 90 ( 
  
 
8 5 B   



 ½ 



  
4
7
   7 1 3 791337

 
 ¾

½

 




REMARKS

'  
 
    
8 5 
 
       

    8 5  
 *!
 8 5%   
 
 !
  8 5 B

'   
    8 5  
   8 5 
 => 


 
 
    ! 
       8 5  

! #
  

4.4.2 Generating deBruijn Sequences
 Æ 
    
8 5 B     

  , 

 
      8 5 
     
 
 
  
&   
 A
    8 5 
    
       # 
 

256

Chapter 4

CONNECTIVITY and TRAVERSABILITY

FACTS

 ' -2 70'      
  !  

# 
  


  

 A
 
' "
8 5 
 =>   B 
 
   

  
      
 


#     
   
 


#  =6  
 
 B    ,
   
 
  
,   
8 5 
>
' 6 B 
 
  
 A
    8 5 

 

 
8 5 B    E 

EXAMPLE

' (    
    
8 5 B  
 8 5 
   



0000

000
0001

1000
1001

001
0010

100
0100

010
0011

0101
1011

011

1100

1010
101

0110

0111

1101
110
1110

111

1111

Figure 4.4.2  $"
 " 

=>

REMARKS

' 6   (
 (    Æ  
 A
   =>
  



# !   !  (
 ( 
 
    
  !   
B 
 
 8 (
 (     ! 
W1%
   
W%
ALGORITHM

' 6 
8 5 B     ( %
  =B

 
 >  
 A
    8 5 
 = > 6  
B 
 
   A
  =( %
 

 
>

Section 4.4

257

DeBruijn Graphs and Sequences

Necklaces and Lyndon Words

( &
 J -(J990 
  

&



  
  8 5 B
DEFINITIONS



'            
 
      
=
   

 B
 
 >



'  B
 
  
     
      

 
    

'  0     
&
   ! 
  
 

 
    :  &
     
  

 + 
&  # 


 
 
    B
 

 

   &

FACTS

'

 &
 

 
:  ! 
    
   

  ! 
  

   !    
  



' 8
 

 
   8  

    
&
   


8= >  




! 8= >        
 - 0 

 

    

'

-(J990' "  =# 
 
 
> 

 
  

 : 
 ! !  



  # 
  
 


 !    
   1

      
8 5
B   
  # 
 
   
REMARK



' 6  - = >  &
 ! #
 !   + 

 - =1> ? 13
 - =4> ? /

-

=4> ? 3 !

EXAMPLES

'

(     
 ,
 B
     4
01101
11010

0

1

1

10101
01011

0

1

10110

Figure 4.4.3

'

 0$

  ""


6  

 :  !     1 + 
 
 11 
8 5 B   

258

Chapter 4

CONNECTIVITY and TRAVERSABILITY

'

6  

 :  !   
 11
 1 + 
 

1111 
8 5 B   

'

+ ! 
 (
 (4  ?
:  !   
 



 " # 
    



111 11 1 1
" ! ! 

  !    
 ! 
  # 
 


 8 5 B
11111111
  

4.4.3 Pseudorandom Numbers
( $ 

 
       
 

 
 = 
 
>
    
  
@       8  

 
  
    $ 

 
  
 B    .8 5 B ! 




!
 
   
   
 
  
  
 
  
 
DEFINITIONS



'   
 
 B 
B   
  

 
 
 
   
 
  



' -2790 6 1
  
   
   B

 !'





6   % 





 @     1% 
    

 
 
     
  
 
  B
 
  


    
   
6         
   
  ! 
5
   = E 

 >
   !   
!
= E 11
   11>





'      

 B 
   
!  '
1 2


 8 5 B   
 
   
# 
   1%
 
   
# 
   %

'

-390 6    .    

 


+

 ,  

= >

7 /
/

! 7=/>        
 
   6  
  
 
  , /     B + 

Section 4.4

259

DeBruijn Graphs and Sequences

FACTS



'

;
  
 8 5 B 
  


  

B



'

 8 5 B    
 , 2% , ! 
 ( 

   % #
 B
    1%    
  ! 
 

    
 B  
   #
 
    
 





'

"
  =
# 
>    
8 5 B   


    
   & 

;  

 
 


 B 

  2 
 5
 

    
 8 5
B  !   ! B
 





'

-H/0' 6 
 !

  

 B   


  


    8 5 B  
 


4.4.4 A Genetics Application
6  
    .F 
 
  # 
  Æ  
   
 
    8
    
 # 
 #     
       
  !& 6      
  


    
  
  
  
.F 
 

 !   6  
  
 # 
  , 
     B    #   
  
  
  !        


         
   
  .F 
B
B   # 

6  
    Æ  
 6



+

 -
6
+
10 

   ,  8 5 
  ! 





   

 B 7

   
 
 B   

 
   
 
 
    
 

   


DEFINITIONS




!)  

'



'

(
  + ?


 ,  B      2 6



7  7¾      7

      



 .F B ! ,  +

 



  

    
    

   +   
  
 

6! 
  
 




5
         
 .F B 7
      !   , 



  
 





   

REMARK



'

  A
 
 
  
 B &       +



 8 5 
 

 
 
   Æ    


!

6   =   
> 
    
 B  

!
  
 #        !  @  

   + 

 8 5 


260

Chapter 4

CONNECTIVITY and TRAVERSABILITY

References
-;0 "
  
  &
 
 &
 :  ! 8 5
B 


 !!! 

M M M&MF&
 M
-390 2 H 
     (     
  N
  
/39
-. 70 F 2 8 5   

  ' 

<
 

=/ 7> 943K97 

/

-(J990 * ( &
 " J :# 
    
 8 5 
B +
 
  %
  =/99> 9K1
-H10  H  H ;       B  /7K1 
8   
  *  
 B!A =   : 8 .

>   L

11
-2
H9/0 $  2

 .  H 
   (   1    
    ' 
 + * (
 X  /9/
-2790  + 2 %  )  % * .
 /79
-2 70 " H 2 F
    
 +
0 $
%
 =/ 7> 79K
9
-22$ ://0  H ( 2 Æ + $ 2
 H * $ 
   :! 
$      + * (
 ///
-2<//0 H : 2
 H <  
    ( 

    
///
-*
790 $ *
 H      8
     /79
-$
10 Q $ &

  
 A ' (    #   

   C %
   , 
  9 =11> //K1
-
6
+
10  
 * 6

 $  +

  A
 



 .F 

  
'

%
/3 =11>

Section 4.5

4.5

261

Hamiltonian Graphs

HAMILTONIAN GRAPHS
' 

' )*   

4 * 
4 6 
  
&
4 A#   
 
4 $ 6
 ; *
 
  T
44 
 2

47 (  



4.5.1 History


   
 
 
 
 F   = -2
H9/0> 
 
 
   
           
! 

 


 
   F
    + 
 !
 *
 
   
       341 *
  #     #
 *

   .   349 6 
 

 , 
  

  
  
      
 
 6 
 !
 
& 
!
 
  34/ 

 !
 
    
   , !

*
 %
    
   
  4 
*
   

    ,  
    B   
 
  "

 -J 470    344 6
   J &
  
B ' 2
  
 
  
 
!
 ,
   = > 


  

# 
   6 J &



&


 B  
 *
  N
  J &
   
 

     
   
 ! (
 

   
   -8 :+ 370
DEFINITIONS

'
'
'

 


 
   


    =
 
>

 


   
   


  


 


  


 
   
 
 
 
   

4.5.2 The Classic Attacks
6
 
 

  
 

      
    

 
    @   
 
  


6

 
&  
  Æ     

   
 


 

         U 
     
 #  

 
  

262

Chapter 4

CONNECTIVITY and TRAVERSABILITY

  
 6   

   
 
   
# 
  (


   
 
     
Degrees

  6      
  


 
#     Æ  = >

 

Æ =

>


DEFINITIONS



' + 

    = >    
   !
 
   6 
 
  

   
 
   
 
    = >



' 6  
      ! = >   
 
  

5   
  
5

  !   
 
      






' (

'  
  

? =
9  > =    ? 9 >   
 
    
 
 
  5  
 
5
 
 ( 

3 9 !   
 
 E 

  6 !  
  
  

  

'

: =

>?


/   (





(½    (

  

FACTS

'
'

-. 40 "
:



  



   

 -;710 " : = >   

 
 Æ = >   

 
 

  : = >    

 -;70 " := >  E  

 
 

 



 
 
 

EXAMPLE

'

  !   !  
#  
  , =
   
(  4> 6  
   
 
  
  * Æ = > ? =*  >

:= > ? *   
   
  . 
% 6
 ;% 6 =(

 > 6 
 
    

 
   !     
 
 
  
 Æ = > ? *
 : = > ? * !  ;  

=
     (  4>

Figure 4.5.1



' -H
310 :

 


2$$"
!  
"  "
&
 ;"& "$


 
   
   !     



Section 4.5

263

Hamiltonian Graphs



' -$$70 "
? =
9  > 


  
  
    =  >
!  => E =>  E   
 
5
 
  
  9  


 


'

-8970 :

! =





   6

>  ! ,

 
 
 
  

 

!½=

!=

>  
 


> 
 
 

 
 
 



' -*/0  

  
  
  
 
 
     
  
 
 

REMARK

'

6   

   
#
       
6   = >
 
 
&   
   ,

 
  
*!
    


      

 !  
"  
   #
 
  
  1   
  
    6 !   
       !    


  
 
  !    

 
   =!    
 
>

Other Counts
DEFINITION



' 6   

# ( 

   - =(>    


 
5
  (     
  - =+ >  
+
     

 
5
  
#  + 

! "   

  +  
  !      ,  
   

  



' 6 
    
 
 
   = >    
  
  ! 
 
        

#




'

 


    



= >  

EXAMPLE

' 6 
 =5 *>  
 
 !   *

#  +

 !
+  ?   ? 5
   ? *  5
 ! !
 

5
     
+       *  
   +   
 
   Y 

 ¾  
  (  4 ! !  
 = 7>
 = 4>

264

Chapter 4

Figure 4.5.2

+"


= 7>

CONNECTIVITY and TRAVERSABILITY



= 4>



FACTS

'



-;70 "


  
   
 
   ½ E  

 

 
 (    
 
 
 !    #
  E 


 = >
 = 4> "
    
  
 
      
 



'






-(
3 0 "

 

/ /
(=> =>  = > ?   


  
  

 
 


' -8
8L: 3/0 " 
  
  

 := >  E = >   
 


 '

-A90 :



  


 
 =

> 

 

= >   = >     


 " = >   = >   
 

 " = >   = > E    
 
 
' -+930 " 
  +    - =+>  
   
 


' -(370 : 
  
    "  #   &  
 

  
   + 
  !  
 
  & ! 
 - =+ >    
 "



'

 
 


-8L/0 -(
2H
:/0 "


  
      

   
 


- =+ >    
  +  !   
  
REMARK

'

6  
    #
 

 1

Powers and Line Graphs

  
    
 !
& 
  
  =
 



>

DEFINITIONS



' 6
   ;= > 

  
 
 !
  
   
    !      
!

 !
   ;= >

5
 
       

5
 =

   
>

Section 4.5

265

Hamiltonian Graphs



'     !  
 
  
  

   

# 

!  




' + 

 

      

 
    5    
 
 = 

 ½     >  
 
  
   
         
  
5
       
' 6    
 
  
 
 !   =  > ?  = >
 !    =  > 
     = >  





'    

 
 
 
  
 
 
 
=
#  5 >      
 
  = >

 " 
 


FACTS



' -*
F+740 : 

 !  

  6 ;= >  
 


    ½       

 
    



' -2* //0 : 

 !   

  6 
 ;= > 

 
 !   =  >   
   

   
  


'




'
'

-+
90 "
-(9 0 "

"
-8930>

  !  Æ = >   


  
 

  





; =

> ? ;=;= >>  
 

 
 


 
 
 =  
 
 
 > =

Planar Graphs
FACTS


'

-630 A
  

 
  
 
  =
 

 
 -6470>



' -2730 : 

 
   !  
 
   !  " 5 
           !
 5¼         # 
 !     =  >=5  5¼ > ? 1



4.5.3 Extending the Classics
Adding Toughness
DEFINITION



' " 

#   + 
   +  ! 


&  
  & 


 , & =+ >  +  ! =+ >    
 & 6     
# 

FACTS

'

-H930 : 
  
  
 
 

     


   
 := >  

 6

266

Chapter 4

'

-8
$L/10 :
 
 


CONNECTIVITY and TRAVERSABILITY


  
  

'

-8L/10 :

  
  
6  
 


 
 := >   6

  !  Æ = > 

  


REMARK

'


G

 5 
  
&  

 &  

 
 

 ( 
 & ?     *!
  -8
8: L110 #
 
=/  6>   
 
 
 
 
 6 $ 1 ! 

More Than Hamiltonian
DEFINITIONS

'
'

 


    
   
   
  2   2

  
  


    

 

    
   
   
  



'  
    
  
 
   !   /  

 # 
    / E  
 
   =! > ( 
  
#

 

#  

   
 

 
  
2




'  
     =
>   
  B  
 
 
  =
 
  > 
  
    B  

 

FACTS

'


  
=> Æ = >  
    => := > 
  


 
 !      
        
    

-8(
2:/90 "



  


  
EXAMPLE



' 6         !        
 

   6 
   
  
    
 
   
  
 

FACTS

'

-8990 "
 
   

'



 
 
  
  

!   = >  ¾    

-*/10 "

   
 := >   
   #
 
    ! 
 
   : = >  =  4> 
  
#
 (  Æ = >  = E >      #


'

-*/0 " ? =
9  > 


  
  
     
 
 
5
 
 ( 
 3 9 ! 
 =(> E =3>  E  

 
  

Section 4.5

267

Hamiltonian Graphs

' -*/0 :  /   " ? =
9  > 


  
  
  
 
   Æ = >  /
  = > $ ¾  / E /¾     
  


' -J
/70 -J
/30 6 # 


    
  


  
 Æ = >   = E > 

    ! 

 
  
 ' -J 
//0


      

:    
 
 
" =>   E     

#        
 

' -(
2J:
0 :  
  !      
  

 
  " => E =>  E =  />  
 
    
5

 
      
 

REMARK

' 8   
 ! 
 
   

   N#
   . 
      !   ;   

4.5.4 More Than One Hamiltonian Cycle?
A Second Hamiltonian Cycle
FACTS

' A
  
 
 
  
 
 
   
 

  6 
  
 
 
 
 


  

 

 
   = -6 70>

' -6/30 "


 
  

 
 



/ 
 !  /  11 







 ! 

 
   !  :  

# 
 
  
  ! 

   !
   '*

  =! > =  -
  =>   =   =! >>> 6 


 
   ! ¼ 

 ! ¼   ? !  
  

#     
    !   ! ¼
      !
      ! 

' -6/90 :



' -*110 (
 
      # 
   =>  
 

 
 
 

!  Æ  = >   => 

 
 Æ = >   Æ   E 

 
   " 
 
 
 
 
 

 =Æ  = >Æ = >> 

 
 
  

' -$
970 -2$
970 6 # 




 !   5  
 
  

! 



Æ  =

>



 

 
 
  

' -Q
970 -3/0 6 #  ,  
 #
  4  

 


= 

 
>  !  
 
  
 
   
 


268

Chapter 4

CONNECTIVITY and TRAVERSABILITY

REMARK

'

(
       6
 -6930 #  %  
 5 


 ! 5   =  
 
 
  ! 

  
  >
6
 #    =(
 >

Many Hamiltonian Cycles
FACTS

'




-6/70 :


! ' (½ 3½  (¾ 3¾    ( 3 (½ 

 
   
 
 

=
> "
 
  3½     3 
 
 
  

   
    (½3½ 



 
 

=> "
 
  3½     3 
   $ 
  ½  ¾     =1  ' 
  >
 
  !       3 ½ (3   

 
 ·½  =  '>V

 
   
  ½
  
 


'

 
 
 


    71¾  
 :¾ = >  E    

-(
340 :

=
> "

   5  
 
  


=> "

   7
  
 
  ¾ ½ E  
 5  
 
  







 



' -A/0 :       !   =  > "


  
!  :¾= > 
 Æ = >     

    5  
 

 

Uniquely Hamiltonian Graphs
DEFINITION

 '

 
  
 
   
 #
  
 
  

FACTS



' -A!310 6 #  ,  
  B 
 
 
 !   
  

'

-H
+3/0   B 
 
 
 


#  
 
= E />
   
 B  
   
 


#   

'

-8H
/30 A
  B 
 
 
   


#  
   ¾ =3 > E  !  ? =  ¾ > ½    ( 
  B

 
 
 
 

 
 !
     
 

Section 4.5

269

Hamiltonian Graphs

Products and Hamiltonian Decompositions
DEFINITIONS



'  
    

        


 
      
   
 
  

 
 
  = E > 


'

A
   !   &    
 

#   = ½>   = ¾>
6       ? ½  ¾ 
  

=

> ? =½ ¾>=½  ¾>  ½ ? ½
 ¾¾

=

6         

=

> ? =½ ¾>=½  ¾>  ½½

?

=

¾>
½
½>

 ¾ ? ¾



¾

 ½½  =

?




  

 ¾ ¾

=

¾>



 ¾ 
  
 = > ? =½ ¾>=½  ¾>  ½ ? ½
 ¾¾ ¾ 
¾ ? ¾
 ½½  = ½ >   ½ ½  = ½ >
 ¾¾  =

6     

½>

½

¾>



6
       =  
      !

> ? ½- ¾0 
  

=

> ? =½  ¾>=½  ¾ >  ½ ½

=

  ½ ? ½
 ¾¾  =

½>

¾>



REMARK

'

H
& -H
9/0 5 
 
  
 
 
   E
  

 
 
  

 B  ' " ½
 ¾
 
 


  

   ½
 ¾
 
 
 
T

FACTS



' -/0 : ½
 ¾  ! 
 

 
  7
 & 
 

   
  !  &  7 6 ½  ¾  
 
 
 
   !  '

7  &
=> &  
=>

=>   

¾

= >   

½

'

 
 


 
 77&  

"  
   
  ½
 ¾
   
      
   *    
 
 
 
   
    
½



'

¾

-8/10 -Q3/0   ½
 ¾
 
 
 
 "


    
    ½ ¾  
 
 


270

Chapter 4

CONNECTIVITY and TRAVERSABILITY

'

-(
: /30 6   
 
 
 
    
 
   ½
 
 
 
 
   
 

'

-8
30 6 # 
    ! 
 
 
 
 

 
 





' -J/90 =
> "

 


=>
=>

  


 
 


 = >    - 0  
 

"   
   
  = >    - 0  
 
 "
 

 
  = >    - 0  
 


=> "
=>

 = >    - 0 

 = >    - 0  
 

"  
  

  = >    - 0  
 

"
 = E > 

 
 

 

  = >   E 
 - 0  
 

"
 

# 
 
   
  = >    - 0

=> "
=>

  
 

 



 

# 
 



4.5.5 Random Graphs
 
  + 
  5= >    
    
 
 !  - ?  
DEFINITIONS

'

=3    
>  
 1  *   :   



  
   
   -    !  
   *
' =3 .  ( 
>  
 4 ? 4 = > 
   


!  
& 
      
  6 
 7 ?   @

 !  4     
#        +     
  
   
 
 !  
   7




'  !
  @

    
    

B
= >
   

 
  

   


 








&   & ? 1     - 
 

' " Z 
  
 
    ! 

  
    
Z 
  "  5=">  
   F 
    B
  
 

   
 
 
   
 
 "   
  



' 6 4 2    %     

 
  


# 
  
  
 2
   
  

 
 
 6    
   
   %    

Section 4.5

271

Hamiltonian Graphs

FACTS

 '

- 970 -J970 6 # 

   

 
 
 
 


 
  ¾   
 

' -J970 -J30  <= >  
  
  * ?   E
 E <= >
 4 = > ?    E  E <= > 6
 




   
 


 


'

 

 
 


4 = > ?  = E 
1 E >'    
  5=    
 
> ? 
'    


'    
' -+/ +/ 0 ( 
 5  

 5 
 

 
 

' -(/ 0 
 
  
 %  
 
 

' -(110 
 
  
 %  
 
 
 " 
 

    
      
  %    
 
 ; 
 


 %   
 %  
  
 

-J30 (



REMARKS

'

"  
   
 
 - V 
  =!  B
 
   >  

  !
 C

D =
  
*> AI
  !
 ,  5 
 !  
       
  
 
 
     
 
&  
 
 
 6  !

 ,  8G




' "  

 
& !  # 
 

  
 !  
       ,

 
       8G
 (
 ( 
-8((340  

  
  
   


'

     B   
    

 

4.5.6 Forbidden Subgraphs
DEFINITION



'  
  
   
  
       

       


   


  =
> 6 
 -   

 
    
 
#  5 

     
  !   
 
 
   
    
=> 6 

=> 6 


;    !
#  5     

  5   
  
# 

   
# 
 =
 


&>

Figure 4.5.3

 !"
 -  
 ;

272

Chapter 4

CONNECTIVITY and TRAVERSABILITY

FACTS

'

-.2H
30 "

 ½  -

=
> 

   

=> 

  

'








 
 

 
 
U
 



-8.J110 6 # 
 
  
   , 

 
  

  -





 


'

-8L/10 "


 


'

-2H
30 "


 

 '

-8/0 "

'

-(
2 /40 "



 
 



 

  

 
 

  -  

  - 

 
 

 
 

  -  


 

 
 

 
 

 
 


 
  



1

Other Forbidden Pairs
 

 B  '      
T 6  !
 
 
  -8/0

 

  -(
2/90  
   1   + ! 
  
  

FACTS

'

,

 





?







+





+

6

 


, + 

-

    


, +



?


 =

  

 -








>




 
 
 


-  

 =

-  


!

1> 
  
     


'

-(
2/90 :

, +

 

, 

'

?

+

9  


?





1 6

  

-2:  0 :

 9  



?

=
> A
  

, + ? >
 
, +    
    -    -  

  
 =

  
  

=>

,

-8/0 -(
2/90 :

  
  

9





'

 9 

?

!

> 

  
 
 
 
   


 


 
  



     
   
 





?



;





-(
2/90 " 
 
 
  



 !    



   

   


 
 


Claw-Free Graphs
" 
     
 

  
!



=

 
   


 6  !  

 B
'

  9
-   -  -   - 
?



6   

   B '



    !  

"  
!  
     


   
    T 6  !

!   

  -(
2H
:10
!
     
  

 ! 




! 

&

  Æ  
  

(  -(
2H
0         
 

    Æ  
 
 ! 
 
 8& -810 
   

     !    
! 
  
      

 " -(
2H
0
   
        
    
 
 ! 


Section 4.5

273

Hamiltonian Graphs

DEFINITIONS




' (

# (  
   
 -- =(>0  

 

 
 (   
 
  
  -- =(>0 


   =- =(>> =;
 


   

!  
  
!
>



' 6
 
   

!  
   2= >  
 


  
 ,   
   

# (      
 



' 6     


    

  

5/=

>    

FACTS

'

-(
2/90 : , +   
 =, + ?  >

  

   1 6
 , +    
   #
 
  
, ? 
 +       -    -  




    -

' -/90 "


'

- /90 :

2=

>  ! ,

=>  

  


5/=

  
 

 
 




!  
 6

=
>  
=>



> ? 5/=2= >>

  
 2=

> ? 

REMARKS

 '

6 
!     @     = -8970> 

 

   
 
   
 (  
  
  -8 110



' 8 (
 73 

 


 
!  

 
 
 
  

2=

> 

'

6


 
 
 
 
 
  
 
  !  
(  
  
 
 
  
   -8930 -8930 -+ 2
3 0
-8/40 -2
/70 -2/0
 -210

References
-8
30 Q 8



 2   
 *
 
     # 
 
 +
 
  %
,  =/3> 4K7
-8
8: L110 . 8
 * H 8
  :
 * H L
 F 
  

  
 
   



$
// =111> 9K
-8
8L: 3/0 . 8
 * H 8
 * H L

  :   
 
  
  *
&

 F 
 +
 
  %
, 9 =/3/> F 
9K 

274

Chapter 4

CONNECTIVITY and TRAVERSABILITY

-8
$L/10 . 8
  $

 A  
 * H L
 :  
 
 !  
      $
9/ =/3/M/1> 4/K91
-8970 H  8
 L 
G

    
  
=/97>   K4

   $


4

-8(
2:/90  8
 2   H (
  H 2
 : :
& ;
     
 
 +
 
    =//9> 74K9
-8/0  8
 -    
      
   . 6  $  
 N
   //

     

-8930 H  8 *
 
 
  % 
 
8 &
  +  A 
   : =/93>
-8 :+ 370 F : 8  A J : 
  H +   
  
N
   ;# =/37>

 
  

:

DE" !E" ;#

-8((340 8 8G
 6 " (
  $ (  ; 
 
 
 

    
 
     $ %
     

  F! <& /34 1K /
-83 0 8 8G
 6 
   
 
 

37 =/3 > 49K9 

 

$
%


-8H
/30 H  8
 8 H
& L   
    B 
 


 +
 
  %
, 9 =//3> 74K94
-8990 H  8 
   


+
 
  %
,  =/99> 31K3



-8930 H  8  *
     

  
   ! %
*


  
  
    
    F SS" =/93> K3
-8/40 H  8  8
  
  K 

  
( A
 
 =//4> 4K1
-8/10 H 8
&

    


     
 J! 
    =//1>

-810 H 8& (     
        $
4 =11> 9K97
-8.J110  8
I
 ( ( .


 A JI : 
  
  

 
   =
! >  
 %($ +


1 =111> 77K
799
-8 110 * 8
 Q  5G
[&
 "     '

 
 
     7 =111> 9K 3
-8L/10 * H 8

 * H L
      
  

     
     ½ ¿  
 
   $   

   8 & A 8" +  L $
  +  Q  =//1> 3K
/ 



Section 4.5

275

Hamiltonian Graphs

-8L/0 * H 8

 * H L
 :  
   
 
 

 !  
     +
 
   4 =//> /K3
-+
90 2 


  A +
 ;  
 
 # 


%
$ 
3 =/9> K 3
-A90 L 
G


  AI
    
 
   
=/9> K

% 

   $




-(/ 0  
  $ (  *
    

  
  

 +
 
  %
, 7 =// > 4K7
-(110  
  $ (  *
     
 

  

 ) %     7 =111> 7/K 1
-2
/70  H 

 H  2

 *
 
  
 
  
 

  


     $
47 =//7> K3
-. 40 2  . 
   

 

=/4> 7/K3


0 $
%




-.2H
30 . .@  H 2
 $  H
 (  

 

 
      

     
   2 


H 
 . 2  : :
&
 . : & =/3> /9K7
-A/0 < A
!
 A  5  
 
    
  ;  
/ =//> 4K41

%7 +
$


-A!310  A 
 * !
 
     
   
 +
 

  %
, / =/31> 1K1/
-(
3 0 2 * (
 F! Æ         

, 9 =/3 > K9

+
 
  %

-(
: /30  (

 H :  *
 
      
  / =//3> 4K44

+
 


-(
2/90  H (

 H 2 

     
  
 

     $
9 =//9> 4K71
-(
2H

0  H (
  H 2
 $  H
 (     

    ' 
 
  
-(
2H
0  H (
  H 2
 $  H
 (     

    '  Æ  
 
  
-(
340  H (
  
   A  5  
 
  
 
   .  

       
 %  =J



$  /3 > +   F! <& =/34> K /
-(
2H
:/0  H (
  H 2 $  H

 : :
& ;

 
   . 
%     $
14 =//> 7K9

276

Chapter 4

CONNECTIVITY and TRAVERSABILITY

-(
2H
:10  H (
 H 2 $  H

 : :
& 

  
  
!     
 
 
    $
 / =11>
9K3
-(
2J:
10  H (
  H 2  J&
 : :
& "   
  
 .       
 
 
 +
 
  
 F  =11> //K1
-(
2 /40  H (
  H 2 Q  5G
[&
 "    ( 


 
     
'
1/ =//4> K
-(9 0 * (  6 B
  
 !  
  
 
 +

 
  %
, 7 =/9 > /K 
-(370  (
  ! Æ      
 +
 
   1 =/37>
14K 1/
-2
H9/0 $  2

 .  H 
   (   1    
    ' 
 (

  F! <& =/9/>
-2: 0  H 2 6 :
\ 
&
 ( 
       
'

    
  
-2/0  H 2 N
   
 
  K

  +
 
   4
=//> K49
-210  H 2 
   
 
  K

   
 
   / F  =11> 9K4
-2* //0  H 2
 A *         
    
 ,

  (


7 =///> 7K 3
-2H
30  H 2
 $  H
 (  

 
 
 
   
    $
 =/3> 3/K/7
-2730 A H 2  
  
    !  
 

   08  $
F  =/73> 4K43
-2$
970 8 2

 H $
&

    5  
 
   
  $
 =/97> /K/7
-*
F+740 ( *


   H  F
 + 
 ; 

 
 



   
 
$
,
=/74> 91K91
-*/10 2 *  A#     
    $
34 =//1> 4/K9
-*/0 2 *  A#      
  
 +
 
  %
, 4
=//> /K
-*110  *
&
 : 
  !      
 
  
   $
 =111> 94K31

Section 4.5

277

Hamiltonian Graphs

-H
9/0 8 H
& A  5  
 
    
 
  
  +

0 $
%
=> / =/9/> K7
-H
310 8 H
& *
     
   
 +
 
 
% , / =/31> 9K 7
-H
+3/0 8 H
&
  + +       
 !   B  

+
 
    =/3/> 499K431
-H930 *  H ; 
# 
     ,  
      $

=/93> /K 
-J 
//0 *  J 
 2 F G
&I
 
  &! ;   
 


 +
 
    =///> 9K4
-J 470 6  J &
 ;  
    
  
     
 ) %  =:>  7 =347> K 3
-J30 H JG

 A G  :         #   
 

    
 
    $
 =/3> 44K7
-J
/70 H JG
 2 F G
&I
 
 A G  ;  B
 

 

    
 )       / =//7> /K
-J
/30 H JG
 2 F G
&I
 
 A G       5
  
 
 

 =//3>  K71
-J970  . J
   
  AI

 G  *
   
    
 %8  $

9 =/97> 971K97 
-J/90 $ J     
 
    # 
   +


0

 =//9> 4K3
-$
970  $
    
   
  

  $
 =/97> 9K 1

 

# 

-$$70 H $
 : $ ; 
 
     
  
 (
+

$
 =/7> 7K74
-;710 ; ;    
 
    
$
$ 79 =/71> 44
-;70 ; ; *
 
  
 +
$
 



 =/7> K9
- 970 : G
 *
 
     
 
    $
 =/97>
4/K7 
-+/0  +  
 F  +
 
   

 
 

) %      =//> 9K4
-+/ 0  +  
 F  +
 
 
 

 
 

 ) %     4 =// > 7K9 

278

Chapter 4

CONNECTIVITY and TRAVERSABILITY

-3/0 $ 
    5  
 
     4  



   $
3 =/3/> 41K44
- /90 Q  5G
[& ;
   
!  
 +
 
  % ,
91 =//9> 9K 
-/90 ( 8  *
     
!  
 +
 
  %
, 4
=//> 9K/ 
-/0   *
      

   
   
$
/1 =//> 7/K/1
-6930  2 6
 *
 
  
  B  
 
 

   $
 =/93> 4/K73
-630  6
    
  

 
 +
 
   9 =/3>
7/K97
-6/70  6
 ;    
 
     
  
 
       
  4 =//7> 9K 
-6/90  6
         
 +
 
  %

, 9 =//9> K 
-6/30  6
 "  
  

 
 
   

 
 +
 
  %
, 9 =//3> 1 K1/
-6 70 + 6 6 ; 
 
    +
0 $
%
 =/ 7> /3K1
-6470 + 6 6    

 
  

$
%
3 =/47>
//K7
-+ 2
3 0 . + 
 H  2

  
 K *
 
    
 

   $
4 =/3 > /K1 
-+930 .  +
  Æ      
 
    +
 

  %
, 4 =/93> 3 K37
-Q
970 H Q
&
  
 
     4  

 
 +
 

  %
,  =/97> 7K
-Q3/0 $ Q .      
   

 
 

       3 =/3/> 43K73

Section 4.6

4.6

279

Traveling Salesman Problems

TRAVELING SALESMAN PROBLEMS

 

  &    +' $

7 6 6
  

 
7 A#
  
7   *  
7 "
 *  
74 6 2
  6
77 6 L    


Introduction
6 6
  

  =6 >  
       
  
   " 
      
 
 6  
  

 Æ  

 


  
 
  6 

 

 
 
 
  #
      
 - 10 "    
!   6   2
  6
  L    

4.6.1 The Traveling Salesman Problem
J $ -$0 !
 
  , 
     6
  


  =6 > * 
 
   
  
  #
  


     
   
   
  
 
 
!
     
 ! 

  
   &
   


   
 

    
    
  
  
    " 

 ! !   !  
     
!  
  !       
 =(
  


!  6     -*+340>
"
 
      

  
    6
 


Symmetric and Asymmetric TSP
DEFINITIONS



' #  3#& =#3#&>'
2

 = > 
  !  !     ,

 

        =
> ! 

'

   3#& =3#&>'

2

   







     ! 

 !  !   
 ,

 
  

280

Chapter 4

CONNECTIVITY and TRAVERSABILITY



' 6 /
 3#&   
 
  6  !  
 
  
  A 
 

  !   
    A 
  
 !
  

'

 
 
   

    

 

  6               
    


 
 




# 
    

  8 3#& !    6
 6  
 
Matrix Representation of TSP

A
 
  6 
 

 !   
 #   !    
   
 

 #      6
  

    6 
DEFINITIONS



' 6    = >   
 
  6   
 # % ? - 0
!    !     !
  
   6      
 
  6   
 # % ? - 0 !    !   
  
         

   
   



'  
  6  
  
    
 
   E   

   
    
EXAMPLES

'

 
  6 !   

1

% ?
9






 #
7 4
1 
1
9 4

1
/
3
1




 !  (  7 6
 V ? 7   
 !  / 9 1  9

 6  
   !        

Figure 4.6.1

'

 
  ,7

 
  6 !   
 
 #
1 1 9
1 1 /
% ?
9 / 1
9 7 /
 4 1





9 
7 4
/ 1
1 7
7 1





Section 4.6

281

Traveling Salesman Problems

 !  (  7     
 
 4
  
 V ?   6
 
      4    !   

Figure 4.6.2

 
  ,,7

Algorithmic Complexity
FACTS



' 6 
 
    

# 
 
  
 
6  
 

  !  
 !'
  !  1  

  U

  !    
           



  
     

 6 

'

(
    
 6  F 
 
   
 B
  

8 
   !  1  
  !    5   

  
 (
  ! 
  !  

'

-
2970 (

 
 
 5  ?F      
  
  

!
 
  
 ! 
  5     


Exact and Approximate Algorithms
DEFINITIONS

'
'

    
  

  

!
 
  
  

    =   > 
  

  
   


&          
      

 

   


  : ! 

# 

   6
 =
 

6 
== >  == > 
== >   !  
  
  
 

 

  
  !  
 = 




' 6 *
   -Q30 

  !  #! =!>   
 =
== >  
== >>= == >  
== >> 
& 

 6 
 =  ! 
 == > ? 
== >
FACT

'

-*
J10 6 
 
  
# 

 

#! =!>   
   6 !  #! =!>   

!  6 ! 

282

Chapter 4

CONNECTIVITY and TRAVERSABILITY

The Euclidean TSP

.  (
 4 !  !

 
  
 
  A 

6  !
   
  
 6  6  !
 ,  
-/30
 //7 = (
 7> $  -$ //0  

  
  

!
 
 = -10>
FACTS

' -
992
2H970 A 
 6  F 

6 $ 1  
 
  
  !" 
 


   A 
 6  ,

   E 6    
   



' -10 ( 


'     !    

 
 =6>> -
/30

!" 
   # 

1=  E


   6

!"
  
      
  
   
 !   6    -10

' -6/90 6 # 

 5 $   
   A 
 6  1= >
  
 A 
 
    , 
 
 
  5   

   
   F 

REMARKS

' 
%  =(
 7> 
  
      
 A 
 
 
 
  *!
 (
 3         
 
 
' A#

  
     
 
  B  
 
 

  =   #
>   
 

   
 
# 

     

   
   !      

 
       
 ( 

 
     
     


 #

 !


' 6    
   
    ! 
'
 
  
   
4
 * *  
   
4  $ 

 

!  6    
    -2340 -H2$<QQ
10

-H$10

4.6.2 Exact Algorithms
6 F *
      
    
 
 
 
  Æ  
 
 
  6   
   F
 
   
 
 
 
 
 !  
 
  !  

 =
   > -8 /30 (
    6 !

   #

 
     -: 10

Section 4.6

283

Traveling Salesman Problems

FACT


' 6  -
   #   #
  
   6   

   
 
    
 
 
     @
 


 =



>V  @  

°





Integer Programming Approaches

L
   
    
 6   
   6  +

 , =-8:110> '*
 
**
=-
30>
  
-''
  
-'-
 = -8
6340 -( :610
 -F
10> 6

 !
&!    
  =-+/30> 6 
  =
   > 

  
   6    .
  (&
 H -.
(H4 0



 
 
      
 ' .,  

 ( 
 ?

(

    

 =  >
1 ! 

:    !  
 =  > 6 6 
  #
'
  > ?









 (
 

5 
( ?   ?      


( ?   ?      


(  +    
 1  +  
¾  ¾
( ? 1     ?     

FACTS

 ' 6 ,   
  

   
#

 #


     
  
 

  
 

# 
#
  6 !   
  
 
 !

5
 


#  
   *!
     
  
 
  
"
  
      
    !  
#  5   
=
 - >

' 6     
 
 - *
  B  

   
 

+

 
 


 

 

+

' 6 
  !       
 
  
 

    
* ! * 
 
  
    1=  > -+/30 
       
   !    
#  5 
 

!     !

 
 

°







"

&

?  
  
    6 
 

284

Chapter 4

CONNECTIVITY and TRAVERSABILITY


 ;!   
   !    ( #
 

 


!   
  
 


  
 

   B  

      6 
  



   

 
   6 

4.6.3 Construction Heuristics
# 

  
        
  

  !   

Greedy-Type Algorithms

6  
  
         
  6 

  
  
 
!
  
  #
# 

   
  
   
 
 
Algorithm 4.6.1: <
" <! " 3<<4

#'   
 
 # - 0

,#
# ½ 
5' 6  ½  ¾      ½ 
" 
  + '?         ½ 
(  ?      
   
  ½  ?   ½  
¾ 
+ '? +   

    
   
   
  


#  5  
   
 

°



 =  > 
  # 
 

°



 =  >

Algorithm 4.6.2: +" #" 3+4

#'   
 
 # - 0
5' 6 =6 > 



+


 =>

 + ? 
 / ? =  > = 6 >  / ? =  > = 6 >
 
 =>
½
¾    
   
    ! 
(  ?       /
" +

  
 =>  
  
#  5  

  
 =>  

+ '? +

 

EXAMPLE

' +
 >>   
  6  A#
  !  
  


 (  7 
  
#    >> 
 
#   
#

  
#  6           !  1 =!    !
>

Section 4.6

285

Traveling Salesman Problems

Figure 4.6.3

 
  ,7


   
 
 #   -H2$<QQ
10  


  
   
 !  
  6  >>   

U  
    
   
 ! 

    

 111]

    
 
 #   6  -H$10
! 
  
 >>  

 !  A 
 

 
  
 6  

    
 >>  6 
Insertion Algorithms

 ,  
        
      

   6  ( 6    
    ! 
    
  
 
  
!
#     ( 6  
 
  ! 
     6
    
 #
 
 !
  6   !   

5 
 B
 !  6 
DEFINITION



' : ! ? ½  ¾        ½  
# B 
    
  


#   !  (

 =
 >    !      
   
 =
 >   
   
 
 =
 > !  
 =
 >
 = > =
(  7 > 6       ! =
  > 6  =
 > ? =  ·½>
    /    ! =
  > ? ½  ¾    
        ½ 
  =
 > ? =  ½>
 ! =
  > ? ½  ¾      ½ 
  

Figure 4.6.4

2"  /"9

" =


 



>

REMARK

'

'*  ,      ,  
    ,  
! 
 , !
   @
  
  # A
   
 !  -^0 
#  

286

Chapter 4

Algorithm 4.6.3:

CONNECTIVITY and TRAVERSABILITY

1"9 2" 3124

#'   
 
 # - 0
5' 6  ½         
: 
   !
       
" 
    ! ?     
( 7 ?     
:  

#     !      -^0
"
# 


 =
  >    ! ?         
 
  !   ! =
    >    
 
  ! =
   > 

 =
 >  !
 
! '? ! =
    >





  2


# 

  !   = ! >    
  



!

 
  = ! > ?    



  

DEFINITIONS

'
'

 

6 
     567 
#  
 
!

6   
     )67 
#   
   
 

   6
  =  ! > ?   = ! >



 

' 6   
     ,67 
#   
   

   ! 

#  6
  =  ! > ? 
# = ! >

 


   6
#       

  B 

!  A 
 6 = -H$10> 
 
 #  !  #  6
 -22<Q
10 ! 
 #     
   A 

Minimum Spanning Tree Heuristics

6
 
        6
 
  6  
-H2$<QQ
1H$10 ( 6      

  
    
   % ** 
 4% :
  <'   
=    7 
 7   -2<//0> + !       ,
  
     ,
  
 %    

 # 

  
!

DEFINITION



' 6 
     
 

    
 


 6 
 
    

  


   "    
      
 !

 


       A 
 6 -H$10
*!
  
   = -H$10>  
 ,
    ! 
              A 
 6 

Section 4.6

287

Traveling Salesman Problems

Algorithm 4.6.4: 6"= #" 36#4

#'   
 
 # - 0
5' 6  ½  ¾     ½ 

( 
   
      
( 
   !   
  4   
  
    
    


 
 
    !     = > ?  = >
4

 
 
  
:
? ½  ¾      ½ =! 

B 
 >
( 7 ?    
"  ?    &  7   ·½       

FACT

' -H
340    
        1= ¿ >
Worst Case Analysis of Heuristics

+  
 
 # 
 
   

      

 

   
    
  6 
  
 
  

  
    Æ 
 $
 

 
  


    
 

  
 
  
   
 

#  ( #
  
 

  ! 2
  6 = 
   44>  
  6  6  


     !
  
 
 
  

 
 
  6   
FACTS

   
 
  6

 

 6 ! 
  
    
 B
   !   >>
 
   B !   
' -2<Q
1
0 ( 


' -90 :  
   # 
 
 =  6 !   

  6    ! 

 
 =  >V  !
 !  
 =    >

 
 =  >V

' -2<Q
1
0 ( 
   ? 7
 
 
  6 ! 

  # 
  
   ! 

 
 =  >V   
 
' ( 6 !  
 B
    %
 
!
 

  
 ! 
 
   
  !     
 !

 4      = -H
340> *!
 
 
  ! 
%    B !   
 
 
  !  

 ! 

 
   V  - $
J
10
REMARKS

'    ,   (
  
    -2<Q
1 0 "  
 



  
    
    

   
 
= -2<1
0>

288

Chapter 4

CONNECTIVITY and TRAVERSABILITY

'

   (
 7    
     (
 4   
 
 @
 
 !
 , 
    L " 


-
970
  
  2 
 < -2<1 0 6      

  !  #    
  
  ?  7 -6 310




   
 

  =  -*
7/0>
   



4.6.4 Improvement Heuristics
# 

  
  
    
 
 =

 
  
    >
 

 
   
 
 
  
 
 
  6  &!  
 


  '
 ,

  ! 
  
  
     =
> ! 
  =
>     


   ( 
  

   
   ! &!

      
 *' 




 * 

       B
 
# 
   6  
   

  6   ! 

     #

   : 
 J 

=+-$ 
 
 
>
   
  
 =-H2$<QQ
10 -H$10>
6
   
  
 
  #

      

     
  " 
 
 ,

    
 :  J 
 
 

  
 
   -210   : 
J 
 
 
 
 
    6  6 
  
 
6 =  -H2$<QQ
10>
DEFINITIONS

'

( 6   8
  
 
  
  
    

 
  !   
5
  !  !    

 = (  74> ;
 
  
    !   6
  

 

 
    =
      #>


Figure 4.6.5



! 

 



 

 2



" "$
  

 2



 

' (     
    

   # 

 

 
 
 



 







 =
>

' 6    
     
     ,

U   @
  
 
   
 =>        

     

  
 =>
      
    
 =


   > 6   


Section 4.6

289

Traveling Salesman Problems

FACT



'

- $
J
10   
   
 
& #
    ,


  
 =    
>  


 ! 






  



1=   > 
  


 6    
 
     


Exponential Neighborhoods
8 




   _=

 >

 
 
  
 


 
   #
 

 #

 
     



 =
>


 '







6    

( 6   
 


    ! 
 #
     !    

     
  
  

  

=

 >

" 
 
  #  6   

!    
     

1= > =1= >>

6

 
    -A; 10 -.+110
 -2<Q
1 0


  

+  
     
   6

#
   
 

  
  !  
 
   
 
       
 
          
 
 
   
     6 

 ! 
  
     6  
#
 ! 
 
 #  ! 

 #
   
 
   
 ! 
  = 
 !
&>  

4.6.5 The Generalized TSP
6

 1' !


 
 
 
       

#   6 -( 
610

DEFINITIONS



'

6

1 
(    3 

 #
  & 
 13#&'

2

!    






  

     

 
  ,

   !    
  #
  =
 
 >
#  
 

 ?     




'

6 


 

  

  
  #
  =
 
 >


#  
   




'

6

 

 

1 
( #  3 

 #
  & 
 1#3#& 


   
 ! 




 



REMARK



'

;
 
  B  W
 
 %
 W#
 %  26


26    !  
 B
  

6 W#
 %
  

26
 26 
 
  
     

  

 
   

290

Chapter 4

CONNECTIVITY and TRAVERSABILITY

Transforming Generalized TSP to TSP
;   !
  
 
   2
  6   
  
6 


6  Æ  

   26  6
 

26  6

     
  -F8/0
 -:
//0 
 

FACTS


'

"  

   -F8/0  26  6    


  
  
 + 
  ,  

  
 6  

 
 
  
    
      
  

    #  6  
 
 
 

  
 


4

  !   
  
 "  
  
 
  
 
!  
 #




 

    
   
  


 

 '

"  

   -:
//0 
26 
 
 6 


! ,

Æ  
  
 
  
  !    
 

  ! 
 



 
 
 
  ( 

#



  ! 

 
  

4
!

 
    

!




4

 
¼

6 !   
 

 
 ! 



!

?

 ! 

½  ½         ½ 
4
¼





¼



¼ 





" 



  ! 

        


 
 
 
   




    26

6 !    
   !    
 ! 


       !    26 
 
 
  
 



    6 
 !  
   ! 

4

    =





4 


> !    
   
 
 
 
 
 

!   8 
  

#



  

¼



  ! 



 
    26 


'

(  

   -:
//0
 -F8/0  
 5  !

*

     
 
    
  
&  



   
  #

 

Exact Algorithms
FACTS

'


 
 #  =-82 <Q
10
 -:
//0> 
 ! 


 

  
  -F8/0
 -:
//0 
    
   
 

  
 
  2
  6  *!
 
 
 
 B 


 
  
     6 
%
 
  

=

> !     
 =>

'

  

 
   26   


  

-( 
610

:


 


  26  
  -F8/0
6 # 

  

5   &   


 

   6

  26 

'

% ? = > 
 

  ½  ¾      

     
% 
  
  
 
  #
 

#  
 ½  ¾        F  
       
  ?       
=  
    
   
   

#   B
 >
-2<10 :

6   &  !

Section 4.6

291

Traveling Salesman Problems

Approximate Algorithms
 

 6    
  #   2
  6     
, 

2
  6 
 
6 

     
  6 


FACTS



'

(  

   -:
//0
 -F8/0  
    


   
     
  
&   

    
 
# 

 



'

6!  

     
  -( 
610
 -82 <

Q
10
    
# 
 = #
>  

6
  

26
 26  6    !  
 
  26 ! 

½      

  
=> :


   
 


 =



  

> 

 



B
   




8




 

8

8

!
½  ¾     

 


 #
 
#  
 



      

 
  
 

    



  


 
 #   26  -82 <Q
10

 
 
  !  @ !   
 



      
 ! 

½  ¾       ½  

# 
 = #
>     6 




 

  
  ! 

 

 ! >

=> ( 
   !   



     

  !   
 

=" -( 
610   

  -82 <Q
10
=> :

 



8

    

 B


*!
  


    -82 <Q
10 ! 
 

 ! 

 

  
  ! 
 
 
     !  

4.6.6 The Vehicle Routing Problem
6


 
! * 4!:

!
   .
 
 


-.

4/0 6   =   
  ! 
  
 
>   
 
 
 
 
 
   6  L        


 

 
    

        
 

    
 
 
 
 
  L  
      
 
  
 
 
) 
  
 
    
6
 

  
 B
 

 

"


   
  






 

DEFINITIONS



'

2

!        
 
 




 


  

1 



 

    
!  !   !
  "
¾  
?  

 
# 1
 
 

"

     


 ! 







1     

65& 



 !  

 
  
!  
 




?  



292

Chapter 4



'

6

CONNECTIVITY and TRAVERSABILITY

  6
 5 & 
  65&'

      
 

E

2

! 


 

 



1   ?

      
 ! 

 "
  ,
L   !   
 ! 
       

REMARKS



'

; 
  
 
  
  
  
  
'

 
 !   
       
  =
#>   
  !  
 

* &'&


 

!  
  @ 

    

 @
  
   - J
+
//0




'

"  
 
     L =

½ >

   F


 
   =  W %> L
  
   -L /70

Exact Algorithms
FACTS



'

6  Æ  #

      L
  
 



  =-8*110 -F
 10 -
J 60>



'

( 
  
   L   
  
  
 #


     

  -6L 16L 10




'

  L 

   6


3 !
"
 -
 





   
   L
  
 
  !  
  -8 10


  

6 #

 

    !  L


 
  6   

  
  
 !  11  

   #

  ! 
  

 
     L
! 
 !
 4
  -
J 66L 10 ; 
 

   L 


   
  
 
 
  
&  #

 

6

  
  
 
  L 

Heuristics for CVRP
L    
   ! 
 '

 
 
L 



 B & U
  
   

  
    




     
 


 

 


6 
 & 
  
  

  

 
@ !  B
 

  
     =-A;0 -2:
10 -6
/0 -6L /30>

(
 L

  
  
 
   B &
 )#     
 
  
   

 
    #

 
     !         
  
 L   '

 
    




&-  


+




REMARK



'

6    
 !
  
   L   



     L !  W 
% 
  W
%
 W
% 
  W%

Section 4.6

293

Traveling Salesman Problems

Savings Heuristics

6 
& +  
     
  
  -+7 0
  &!
     L  * !  
  
    ! 

  
   -2
/0 -+7 0
 -:
10

  =
> (

#  +  &=+ >  =

# 
  >  !  
  
 6      
  + 
=> 6 
 
 

#  +  =+ > ?

 
¾



DEFINITIONS



'       !½
 !¾  =!½ !¾> 
  !
#
 B
  =!½>
 =!¾> 6            
  !½
 !¾ 
      
 
   
   
# 

  " =  = =!½>
 =!¾>>  ">



' 2
   !½
 !¾   
  =!½ !¾>  7=!½  !¾> 

  7=!½  !¾> ? &= =!½>> E &= =!¾>>  &= =!½>
 =!¾>>

°

'

: , ? !½  !¾     ! 
   /     ! 
!  
 

# 1 6 
   %=,>   !   
  /

  
 !½ !¾     !   

 =!  ! > #   = =!½>
 =!¾>>  "
  ! 
  
 =!  ! >   
 7=!  ! >
REMARKS

'

"   "-( 
 
 
 * =-+7 0-:
10>  !    

!  

  
  &= =! >> 6 
  #

  &= =! >>  !

  

6     
   =! > !  
  


 
 



' 6   !
    !½
 !¾       
& + 
  " = 1>  
  !½ 
 
# 1
 =1  >  
  !¾ 



# 1   !

 

 =  > 
   
!   = (  77>

Figure 4.6.6

 6$
"0>"! '"!  $ !½
 !¾

REMARKS



' 6 
  !
  4     

 =!  !
> !  
#
 
 7=!  !
>
  
  4  " 
   !     
  

 4    
 
 -:
10

294

Chapter 4

CONNECTIVITY and TRAVERSABILITY

Algorithm 4.6.5: ,
/! #" 3,#4

#'  
 
 # - 0 
   ?      


  "
  
  

5' L  , ? !½     ! 

" 
     / ?
   ! ? 1  1  ?      /
" 
  , ? !½     ! 
+  / $ 

 
  
 %=,>



  4  %=,> !  4   /  
( 

 =! ! >  4 
, '? =,  !  ! >
=! ! >

/ '? /  

' ;
 ,   
   
      ,  


  6 
    =   7 > ( #
  -:
10
L     
& + 
   
 
   

    
    
Insertion Heuristics

" L 
 
  -:
10 ! 
        ! ? 1   1
6
       
      B
  

 

6 ! W

%  
 


#          !
 !       ,
 "  B
  ! 
  
!    !  
  
  
 
   

 

 6     

#  
   ! 
  

 &= =! >
>  &= =! >>
REMARKS

'  #
  

     , $  6  
   -$ 69/0
' ( 
 H
&
 -( H
30   

 
  
  
6
   1' 
* ! *  , W 
%   
 


  

  

Two-phase Heuristics

6 
  
 &-  
   
  
          
½      
 
 6  
   
   
1  ?      
+
 * 
-+*90 
!   B   A 
 L 
 !    1

      
     A 
 


   A 
  
 ! 
  

+ 

  
    !    
  
  
     !  
  
 =1 > 6 

#  ?  

 
 =8   > ! 8  
 !  
  1  
  1 
 6 ! 
       
 !

Section 4.6

295

Traveling Salesman Problems

Algorithm 4.6.6: ,:! #"

#'  
 
 # - 0 

 8      

   ?       

  "
  
  
5' L       



!



?      

 
  ½  ¾       
 8

" 
  + ?   ?      




? 

(

7

?     

" =+

  >

'?









 

8
½

7

?       

$ "

E

 '? +
 
?      

+

(



:

 
6    
   +

!

1

REMARK

'  #    

   
 L     -8 /40
 #
 
! 
    

 

  
 
-*$ 69/0

References
-A; 10  J 5
 ; A H 8 ; 
     
 

 
 
   
  B    


$
 =11>
94K1
-2
/0 J  &
 8 2


 
 
      

 5

)
/ =//> 47K 7/
-8 /30 . 
  A 8 #  L 
G


 + & ;    
6
  

  
$
 +
$/ *6  /
($ ,   !!9 (((
=//3> 7 4K747 6 M4M/// 
      



 



  

-/30  
  
  
# 
    A 
 6
 
   +$ 4 =//3> 94K93   

    9
"AAA   (    //7
  3 "AAA   (
   //9
-10  
 # 
 
     6  "   8  %
    /   =2 2 
    A> J! 11
-8
6340 A 8


  6 8

 8 $ "   8  %
 1         5
  >  =A : :
! H J
:
  * 2   J

 . 8   A> +   /34

296

Chapter 4

CONNECTIVITY and TRAVERSABILITY

-8:110  8
  :
  
#
 
    
  


    ( 
%
$

  111
-82 <Q
10 . 8   2 2  $   <
  Q

 6


   2
  6  6 ' # 

  
  
5

)
0
 =11> 49K74
-8*110 N 8

 + *I
  
    

  
 
     G  H
 . (   JI
 6
 111K37
-8 /40 H 8

 .   :
  
  
     
  
 5

)
 =//4> 7 /K771
-8 10 H 8

 .   :
  
  8
     



 L    /  )   =  6
 . L  A> "$
11
-+7 0 2 
&
 H + +    
  

  
  
   5

)
 =/7 > 473K43
-$ 69/0 F  ,  $  
  6 6 L    
"    5
  >  = $    6
  
  A> +  
/9/
-.
(H4 0 2 8 .
  .  (&
  $ H    
 


  

  5

)
 =/4 > /K 1
-.

4/0 2 8 .
 
  * 
 6 &  
   $

%
7 =/4/> 31K/
-.+110 L 2 . &
 2 H +     #
  
  
  

 
  B

 
   $

 
 %
 39 =111> 4/K4 
-A;0 ; A H 8 ; 
   (
 
 
 
 
  
   
     
'
  
   
 $
  11
-( :610 $ (    : 
  6 A#
 $     
6
  

     8  %    /  
=2 2 
    A> J! 11
-( 
610 $ (   H H 


 2G

  6 6 2
  6
 



 ;       8  %   
/   =2 2 
    A> J! 11
-( H
30 $ : ( 
  H
&
  
 
     
 
  '.   =/3> 1/K 
-2
2H970 $  2
   : 2


 .  H  F  
    
9 $ %
  

=/97> 1K

Section 4.6

Traveling Salesman Problems

297

-2:
10 $ 2
 2 :

 H < 
 $
     



 L  "  /  )   =  6
 . L  A> "$
11
-22<Q
10 ( 2
 2 2   <
  Q

     
 
   6  *
+
5

)
/ =11> 444K473
-2340 8 : 2
 +  !
 A  
 
    *    
 8  % 1         5
  >  =A :
:
! H J :
 *2   J

 .8   A> +   /34
-2<//0 H : 2
 H <  
    ( 

    
///
-2<1
0 2 2 
  <  
  5

)
0
1 =11> /9K//
-2<1 0 2 2 
  <  

# 
 
    6
  O !  

  
      


$
/ =11>
19K7
-2<10 2 2 
  <    

 
 
 

  2
  6     +
   9 =11>  /K4 
-2<Q
1
0 2 2   <
  Q

 6
  

   
 '  
 

            6     



$
9 =11> 3K37
-2<Q
1 0 2 2   <
  Q

 A#
 F 

. 
  
     6     8  %    / 
  =2 2 
    A> J! 11
-*
J10  *
 
  J > # 
  +
   =11> /K

-*
7/0 ( *

   
     +  /7/
-*+340  H *@

  + *      8  % 1 
       5
  >  =A : :
! H J :
 *2
  J

 .8   A> +   /34
-H2$<QQ
10 .  H 2 2  :  $2  < + Q

  Q

 A# 
 
    *    6     8 
%    /   =2 2 
    A> J!
. 11
-H$10 .  H
 :  $2 A# 
 
    *   
6     8  %    /   =2 2 
  
 A> J! 11

298

Chapter 4

CONNECTIVITY and TRAVERSABILITY

-H
340 .  H
  *

    
 

    
   8  % 1         5
  > 
=A : :
! H J :
 *2   J

 .8   A> +  
/34
-:
//0 2 :

 (  
 
 

  


  
      
  
  

  ('-5) 9 =///>
 K1
-:
10 2 :

 (  
 
 *     

 
 L  
 /  )   =  6
 . L  A> "$ 11
-: 10  :
    6 !
 "   8  % 
  /   =2 2 
    A> J! 11
-$0 J $ .
  *   *  $  ; 
 =/> K
-$ //0 H  8 $  2   
 
# 
  
 
 '
    
  
# 
      6   $6


  %($ +


3 =///> /3K1/
-F
10 . F
  
 6
 8

       
  6     8  %    /   =2 2 
  
 A> J! 11
-F
 10 . F

 2  
  8

      

 

L    /  )   =  6
 . L  A> "$ 11
-F8/0  A F
 H  8
  :


 


  
  

  
  

  5

)
/ =//> 7K7
-F8/0  A F
 H  8
  Æ  

    
 

  

  ('-5)  =//> /K 
-
990  *

    6 A 
 
  

   F 
 


%
=/99> 9K 
-
30  *

   
 J       5
  >   
*
 /3
- J
+
//0   2 J

   $ +

  
 
 
 
  
     A
 N
  
 A 
"   A //3M
- 10    6 6
  

 '  
  (
 

L

     8  %    /   =2 2 
 
  A> J! 11
- $
J
10    ( $

  J

  6   '  
 

  
 #      4 =11> K9

Section 4.6

299

Traveling Salesman Problems

-
J 60 6 J 
 : J
 +  
&
 : A 6 ; 


 

     

  $
 

-
/30  

 +   # 
    


C
D

C

D 
EA 
$ %
  

=//3> 4 1K441
-210  
 ( 2
 :
 

 $
      8 
%    /   =2 2 
    A> J!
11
-90 L "  &  A 
  

     6
  


  '  $   
  
=/9>
3K =  
>
-
2970  

 6 2
 
# 
   +$ 
=/97> 444K474
-
970 L " 


 ;     
  
 
  
 
 
  / 
'8 ,%%) %
- >
$
'8  =/97> 9K = 

>
-6
/0 A 6



 

 
  
     '
.   =//> 77K79
-6 310 6 + 6   *
 
    
%
, / =/31> 73K9 

¾ /  3 +
 
 

-6L 10  6
 . L  8

 8     

 
 L 
"  /  )   =  6
 . L  A> "$ 11
-6L 10  6
 . L  $ 
#
 
 #


   




        


$
 =11> 39K4
-6/90 : 6

 + *
   A ' 
# 
     
6
 $6 
=! $ %
  

=//9> K/
-L /70 . L     
   
   

 

   
 * 
+
5

)
3/ =//7> 13K7
-+/30 :  +  (    +   //3
-+*90  +
  * 
    
     
 
  
   5

)
I 
 =/9> K 
-Q30 A Q $
   B
  
# 
      

   $
5

)
7 =/3> /K

300

4.7

Chapter 4

CONNECTIVITY and TRAVERSABILITY

FURTHER TOPICS IN CONNECTIVITY
  
 

     

 
  

     

9 *  

9 8 

9  
 
 
9 2
 
    





Introduction
      
   
 
    ! 

  =Æ >     ! 

   
 


 

  
  (  !      
# 
 =
#   >

  F# ! 
 !      
 =
 !>   

 

 ( 
    
 
 
 
 

    
 C  
 
 D
  
( #
     
  


4.7.1 High Connectivity
  
 
   !  C D  
 ! 
 #  ,
  
 !  C   D  
 . @  
   !   
    

'
=
> L  !    
=> 
 
 = 
  >
=> 
  
  = 
 
  >
=> :
  
  = 
 
 
>
6       
 

     

   
!  
# 
#   
 


Minimum Degree and Diameter

  : ? =  > 

 !        Æ   



 =
# >
  "        

  Æ  Æ    

  
  Æ  `

  
 
DEFINITIONS



' 6   

 ! 
          
  

 
 ,   

Section 4.7

'
'

301

Further Topics in Connectivity

6   


 
# 7& = >
¾

6
  



  
 
  



 

<=

>   
#   

 *   
#  
  
    * 

'  = >

= 
> 
FACTS

'
'

-:9


? Æ

'
'
'
'
'

Æ       

B
      Æ 
0 " 
 
5

  
  => E =>    

-770 "

% ?  
? Æ 
-L330 "   
 
 Æ     E  
? Æ 

-L3/0 "  * 
  =*  >
    
½ Æ    
? Æ 
 
-6L/0 "  * 
  =*  >
 Æ  
    ? Æ 

-.
L/40 " 
  B  <  *
    
Æ    
? Æ 
- 940 "



 !  


REMARKS

'
'
'

 (
   (
 

  
  (
 4  * ? 

  
  ! (
 
(


"
    (
 9 
  -.
L/40 
  Æ     
B
    

      
 !   
 
    - Q3/0
Degree Sequence

  (  #    

# 
 !   B
      ? Æ  (

#  - =>    
 
5
  
FACTS



' -2+930 " 
#  

  
      
 
 
=  > =
     C
D
# >  
 = > E = >  
 ?           

? Æ 


'



 '



 
 ,
   
     
    4
=>  
   9
   4

-2A9/0 " 

#


? Æ

¾







-89/0 :


 !     "   B    
  ? Æ 
 ,  = E  >     
  !          Æ 

? Æ 



302

Chapter 4

'

-.
L/90 " Æ      Æ    

Æ      !      Æ  
? Æ 

CONNECTIVITY and TRAVERSABILITY





= E  Æ

'

-L10  


   7 !   B 
!  

 ? ? 1 !   " Æ      Æ
Æ
  #  Æ  
? Æ 
    =Æ E > 

 # 



>  =  > E

<  * : ? ? 
    


REMARKS

'

F 
 (
 3   (
   !  
 (
 /
   (
 
$

 !   #
  - Q3/0 (
 /    (
 

(
 



' (
 1   (
  !  
   

 !  - Q3/0  
  (
   
 /

'

(
   

    


  S -S/ 0 ! 
  =
(
 >  
  ! (
    (
 1



' (
  
    -L330 -L3/0
 !
 (
 9 (
 !  -*L10      (
 
 

 
#  
 

 
 
 
  
   U 
 
=> ?  => =>
Distance
DEFINITIONS



' 6    7& =  > ! ! 
    
      
 7& =  > 

   
 
  
    !      >

 = >  
  =+



' 6
   ; 

 

       

!
 

5
 
      

5
 =
 
 
     >
FACTS

'

:  
     

 
   ?     ?  
6   
 !   
   ; 
 , $=    > ?
 =  > E 
   
  ;
 
  %=; >  %= > E 

'

- Q3/0 :

'

-8

(
( /70 :


 
  
 
 
 
# 
 
# 
  

    
 
  ! 
 , 7&=   >   6
=
? Æ >
6

;
" ;



 !     

=
> "


 

   

=>


 
 ! 

Æ

   


;

 
# 
   =
? Æ >

 
# 
  = ? Æ >



Section 4.7

303

Further Topics in Connectivity

REMARKS

'

6 Æ     
  (
  
#     
  (
 
(  Æ  B  
        


     
 !  (
 4=
>


'

( 

 
& (
 4=
> 
   (
 
 (
  =  &%
>

Super Edge-Connectivity

* !  
 
   
 
DEFINITION



'  
# 
   
   
 
      
     

U 
         

#    

EXAMPLE

'

(  9 !
 
 
# 
   
 
   


 6     
 

        
e

g

Figure 4.7.1

f

 '
9'
$$ !   "


FACTS

'

-:9 0 :

=>   

'

 ¾  ¾  " 
 
5

 
?
 


" 
 
5

 



'

'

-J90 "

Æ    E  


  => E


  => E =>  E  



 


-( /0 " 
 
 !
 
   
   Æ
 
!   => ? Æ 
   = >   


 '

-/0 :





 !  
#   ` "

$ Æ E `   



REMARKS

 '

(
 9
 3 ! 


  (
 
 
   B
 (
 7

304

Chapter 4

CONNECTIVITY and TRAVERSABILITY

'

(
 / 
  

,  (
  =!   
   
 B >
 
 (
 1


 

Digraphs

   
    
 

 


** 
'

=
  
  
  

  
  
      


B
   


 > 
  

DEFINITIONS



' 6
    
 
 
   = >        

#  !    
    





' 6   
 
 
  
= >        
   !    
    


  = >  =>   -'
  

 
5
  *
# 
   =>   -'
   

 
5
 
# U Æ => ?   =>   =>
=> Æ  ?   ¾  =>
 Æ   ?   ¾   =>
=> Æ ?   ¾ Æ => ?  Æ   Æ  
  =
> (

#






  
 
  !  ` 
  
#  

   =
    


  (
 

 5 

>


=  >

  
#   

FACTS

'
'

% ?  
? Æ 
-S/ 0 :

 
    " 
   
  = @>

  =   >  
 Æ = > E Æ = >    ?        
? Æ 
' -*L10 : 
 
 !  

  ! 6
=> ?
  =>  => 
 
 
  
   
' -*L1
0 : 
   
 !   



    Æ  " 
 
# 
 
 
#  
 9
  
  
# 
 

#    
  
9  
? Æ 
' -*L10 : 
* 
   
   
   
 Æ ! 
*   "  =*Æ >=*  >   
=> ?   =>  => 
 
 
  
   
' -*L10 : ¼  ¼¼
 
   
  
     Æ  
!    
   
  " =(> E =3>  = E >  
 
 
 
( 3  ¼
 
 
 
  ( 3  ¼¼ 
=> ?   =>  =>

 
 
  
   
-H90 "





 
 !  





REMARKS

'

F  
 Ga&%  =(
 >   

B    
 H
 =(
 >   
  (
  
  (
 3

Section 4.7

305

Further Topics in Connectivity

'

(
  !
 
  .
&

 L&
  ! B 

-.
L/9.
L110 !   
  
 !

  



'  
  (
  
 

 
 !  
 ! 
 
# 

  
  $
   
   H
%  =(
 >

   
 
     
 
  -(;!110

'

 B  (
     
   (
  

Oriented Graphs
DEFINITIONS



'   
   
 
            
     

# !        
     

          
 
5
   

# ! 
   



'      =
 

     > 
 

 
 !
 !
     
   = >  == > 
= >>
EXAMPLE

'

(  9 !
 
 
# 
   
 
    
"  ? ( 3        = 
    - 0

       >
    

 =       
 
5
   

# !     >

u

x

v

Figure 4.7.2

y

 '
9'
$$    " 

FACTS

'

- (910 :


   
 ! 

Æ  = E >  
? Æ 


 
     Æ  "



' -( /0 " 
   
 ! 
 
    
   



'

-( /0 "


   
 !  
 ! 

Æ     E

 


306

Chapter 4

CONNECTIVITY and TRAVERSABILITY

REMARKS

'

(
 9
 3


  (
  !
 (
 /   
  (
  
B  (
 /



' " 
  Æ     
  - (910
 -( /0 =(
 9
 3>
! Æ  E Æ   
 Æ  E Æ    E  
  (   
 
! 
 (
 3
 /    
 

'

*  
   * ! 
    
   
 
   

Semigirth

6 
  H
%  =(
 >
 
 !   
   

   
! 

 
   
     
 "
 #   

 

   
 =
    
>     


DEFINITIONS



' -(
( 3/( (
A/10 (

  
 ? =  > !  
 %  
   @= >   
  @ ! 
 %  
 
    
=
>  7&= >  @       !
&   B
 
 
  !
&   7&= > E 



7&= > ? @          !
&
'   
 
 
( * 
 !  
 
#  
   

=> 

* 
   
 

    


5
  
   
# 
 6

       


 
   


EXAMPLE

'

(  9 !
 
  
  !     

 
  @ ? % ? 

Figure 4.7.3

,'!" @ ? % ? 

@  B
  

Section 4.7

307

Further Topics in Connectivity

FACTS

 '

-(
( 3/0 :

'

-(
( /7
 8
210 :


 
 !     

@
 
 


=
> " %  @ 
? Æ 
=> " %  @     

 ? Æ 
=> " %  @      



 

Æ $  
 %   

*   =*  >

%  @ E *   
? Æ 
" %  @ E *