handbook of graph theory

CRC PR ESSBoca Raton London New York Washington, D.C.

H A N D B O O K O F

GRAPH THEORYEDITED BY

JONATHAN L. GROSSJAY YELLEN

DISCRETE MATHEMATICS AND ITS APPLICATIONSSeries Editor KENNETH H. ROSEN

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 ORourke, 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 Applicationswith Maple

Patrick Knupp and Kambiz Salari, Verification of Computer Codes in Computational Scienceand Engineering

Donald L. Kreher and Douglas R. Stinson, Combinatorial Algorithms: Generation Enumerationand 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 CrytographyRichard A. Mollin, Quadratics

Series Editor

Kenneth H. Rosen, Ph.D.AT&T Laboratories

Middletown, New Jersey

and

DISCRETEMATHEMATICSITS APPLICATIONS

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 andCoding Design

Lawrence C. Washington, Elliptic Curves: Number Theory and Cryptography

This book conpermission, anreliable data aor for the cons

Neither this boincluding phopermission in

All rights reseclients, may beCenter, 222 RISBN 1-58488a photocopy li

The consent oor for resale. S

Direct all inqu

Trademark N

identication a

Hand

In IS 1

QA 511

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

tains information obtained from authentic and highly regarded sources. Reprinted material is quoted withd sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publishnd information, but the author and the publisher cannot assume responsibility for the validity of all materialsequences of their use.

ok nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical,tocopying, microlming, and recording, or by any information storage or retrieval system, without priorwriting from the publisher.

rved. Authorization to photocopy items for internal or personal use, or the personal or internal use of specic granted by CRC Press LLC, provided that $1.50 per page photocopied is paid directly to Copyright Clearance

osewood Drive, Danvers, MA 01923 USA. The fee code for users of the Transactional Reporting Service is-090-2/04/$0.00+$1.50. The fee is subject to change without notice. For organizations that have been grantedcense by the CCC, a separate system of payment has been arranged.

f CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works,pecic permission must be obtained in writing from CRC Press LLC for such copying.

iries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431.

otice: Product or corporate names may be trademarks or registered trademarks, and are used only fornd explanation, without intent to infringe.

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

No claim to original U.S. Government worksInternational Standard Book Number 1-58488-090-2

Library of Congress Card Number 2003065270Printed in the United States of America 1 2 3 4 5 6 7 8 9 0

Printed on acid-free paper

Library of Congress Cataloging-in-Publication Data

book of graph theory / editors-in-chief, Jonathan L. Gross, Jay Yellen. p. cm. (Discrete mathematics and its applications)

cludes bibliographical references and index.BN 1-58488-090-2 (alk. paper). Graph theoryHandbooks, manuals, etc. I. Gross, Jonathan L. II. Yellen, Jay. 166.H36 2003'.5dc22 2003065270

PREFACE

!

"

"

#

$

% &

Format' (

)

$ * ( ) +

(

" $ ,- +

. /

! ) (

. 0

'

)

'

!

1

( "

Terminology and Notations2

3 4

. )

(

5

0 (

6

'

7

' )

8

/

7 "

) )

*

' 2(

9 ( )

: 6

7 *

( ;

6 )

2 ) .

Feedback" (

(

Acknowledgements+ ( ( 5

? ( 0 ? ? 0

? ( 5 "

About the Editors? 2

? ;

0

CONTRIBUTORS

0 A 0

#

5 0

# ! $

> 0

# ! %

@

+ 5(

& # ' (

5D 5

D

# !

$ $ #)

5 5

# !

5

# ! '

1 5 (

# ! *$ #)

2

( #

?

+ , #

>

# ! -. /01$

0 E >

# ! ($

>* +

2

3 &4 '22

.$ /0

= >

# ! $

? /F

# ! $ /

/

# !

' /

# ! - %1$

= 0 /

# ! $ /

@ /

5

#

8 /

# ! %$

0 /$

#

8 9 2

# !

=( 2

&

? 2

3 #

? @ 2

#

2 2

$ # ! .$ #)

" + 8

3 / #

= 0 8

# ! *$ / !

2

/ '

# ! - %1$

? E

# ! /$

? @

# ! $

2

@

# ! &$

5 =

/

/ ==

!

CONTENTSPreface1. INTRODUCTION to GRAPHS 1

2. GRAPH REPRESENTATION 56

! "# $%

&& "#

' &( &

!

3. DIRECTED GRAPHS 126 )& * "

& +&&& '

"#

$

$

%

4. CONNECTIVITY and TRAVERSABILITY 193' &(, " -& '

# %& %

' . '

' %(

' " "#

'' )/ -0&

$ )

' $

'$ ( - "#

' & &(

# %& %

5. COLORINGS and RELATED TOPICS 340 '

* +

& $

* +

! - 0 %

' & &1 '

( ,,

"& '

(

$ +& # ''

-, ),. / $

'

6. ALGEBRAIC GRAPH THEORY 484$ + '%

-

$

#(

$ . $

$ "(,

$' 2& -&

$ 0$ 1 ,

$ -&

( )

$$ * * '

, 23

7. TOPOLOGICAL GRAPH THEORY 610 -& $

,4+ ,4+ 4(

* *3 !# $

# $'

' 2 $$

$%'

, (

$ * $$

(

(

) ((

%

" 5,

$

8. ANALYTIC GRAPH THEORY 787% .3 %%

6 6 , 5!

% %

5( ,

% %

# %

%' "##& * %$

%+

%$%

9. GRAPHICAL MEASUREMENT 872 & %

*

%%

' ( '

&

% (

' )4

,

'

10. GRAPHS in COMPUTER SCIENCE 952 -&

' 5

& + %

, ),( 7 %(( ## % 7

4

## ,

' + &( & '$

!

$

11. NETWORKS and FLOWS 1074 *3 4

/ "

* 4 %

%(

*& +

) "

' & 546 *

( )! ",(8!

Chapter 1INTRODUCTION 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 INTRODUCTION TO GRAPHS

1.1 FUNDAMENTALS OF GRAPH THEORY

Introduction

!" " "

#

" " #" $

1.1.1 Graphs and Digraphs% #& # #

" ' " " (

$

# #

Basic Terminology

DEFINITIONS

) % * +,

+ ,

- "

%

#

+ +, +,,

./ /

) ' . " # "

) % . . & #

) & #

Section 1.1 Fundamentals of Graph Theory 3

)

) % &

) %

) % # .

#

) # #

#

) % &

% 0/1 01 #

. 01

' " 01

" /

EXAMPLE

) % * +, 2 ' ./

* / * /

" /

Figure 1.1.1

REMARKS

) % 3 / "

" "

4 "

$ "

) (" "% /

56 7 +. /" 6

,

Simple Graphs8 " #

# # # #

DEFINITIONS

) % / /

) % .


) % ./ /

Edge Notation for Simple Adjacencies and for Multi-edges % & # # &./

(" + ,

" # / ."

" # " 2

#

EXAMPLE

) 2 / *

Figure 1.1.2

General Graphs8 9 / 8" /

"

+ /, + :,

% 01 / /

" " "

DEFINITIONS

) %

/ +' /,

)

&

)

. /

EXAMPLES

) 2 #3

4

Figure 1.1.3


)

2

Figure 1.1.4

) / 2 ;

Figure 1.1.5 ! " !#!"

Attributes% # # #

#" # # "

$ +" , +" ,

% # + , $

;

DEFINITIONS

) % ./ #

#

) % / #

#

Digraphs% # %/

! & $


' "

) %

) % + ,

) % / /

) % + , #

' . " 01

)

+" /,

Ordered-Pair Representation of Arcs ' " . .

+ , + , = / #"

+' $ & , 2

#&/ " /

> #& /

& ? #

!

/ !

$ +"

, = # " /

# # - "

# " #

"

9 > #&

EXAMPLES

) 2 @

/)

Figure 1.1.6 " !$


) % #

Figure 1.1.7 % !$ "

Vertex-Coloring= ./ "

$ ;

DEFINITIONS

) % ./

$

) % ./ & >

) % ./

) +, " +,"

# /#

REMARK

) " "

"

+," # # 01 /

# 01

EXAMPLE

) 2 A / ./

/#" # %" #

//# //#B "

+, *

Figure 1.1.8 %" +, * +, *

1.1.2 Degree and Distance

. $ # C


Degree

DEFINITIONS

) + , . " +,"

# # / +2

" " # #,

%

" .

) 9 # .

/

) . # B

. # - /

) % . 6

EXAMPLES

) 2 C 9

6" " " " @" @"

. 3

Figure 1.1.9 " #!&

) 2 6

.

Figure 1.1.10 !" " '"&

FACTS2 " 5DCC" 7 # .

() +-, #

() ' " # #

() % /

9


() ' " # 9

#

() 9 " / 9

() " /" 9

9 " #

Walks, Trails, and Paths

DEFINITIONS

) % 9 "

*

* "

'"

"

"

' " ! # # 9

) *

* "

&

+ ,

% .

) # + ,

) % ! . .B "

) % !

) % ! ($

+ ,

) % .

) %

) % !" " .

EXAMPLE

) ' 2 " . 9

! " . 9

Figure 1.1.11


Distance and Connectivity

)

! #

) . .

!

) % # !

) %

) % . .

!

) . .

)

) .

EXAMPLE

) 2 B

#

Figure 1.1.12 "$ &&" %)$ &&"

1.1.3 Basic Structural Concepts= # 9 "

" #

Isomorphism' " "$ "

% .//. //

"

DEFINITIONS

) % . #&

)

" . & .

+, & +, '"

#&

+, +,


) % #&/

)

)

"

& #&

& +, +,

) =

*

) % .

. !

!

+ , *

&

6

) %

+ ,

EXAMPLES

) 2

Figure 1.1.13 % & ' $ .

#

9/ ! " 0! & !1

& )

!

*

6 6

6

6

6 6

!

*

6 6

6

6 6

6

= #

. !

. !


) 2 " ! 9

> ./#

Figure 1.1.14 % & "" ) #!" *"

) 2 ; / 9

+' ,

Figure 1.1.15 % % #!& "

FACTS

() # #& ./ /. /

9 "+E,

() % ! + ," ! /

() # " # " 9

( "

" # -. ;

() - + , & . 9

.

Automorphisms 3

DEFINITIONS

) %

)

+, *

)

+, *

) % #

) % #


FACTS

() . # ./

() # /

EXAMPLE

) 2 2 @" . #

" #

./ /

Figure 1.1.16

Subgraphs

DEFINITIONS

) % #

+F" # # # ,

) ' " *

"

+ ," ./"

"

++ ,, * ++ ,, * +,

) % # +, * +, +%"

# " ,

) % # #

' " (

#

EXAMPLE

) 2 2 :"

# #

#"

# # #

Figure 1.1.17 !

!& !


FACTS

() ? ) # " # # #

# # +#,

() ' # # # # "

*

2

Graph Operations

" "" # "

# $ "

DEFINITIONS

)

* +," "

. "

+ . #,

) * +,

. #

)

+ , * +, &

. + ,"

+ ,

) + , * +,

+ ,

) % + , . #

) % #

) + /

, . "

) + , # G

+G, * +, +,

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

) + , #

+, * +, +,

+, * +, +, +, +,

+ , +, +, + , + ,"

+, + ,

+,+, + $, + ," $

+,


) ./

/ & " " ./ /

) % & %

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


Acyclic Graphs

DEFINITIONS

) % +" $$$,

) % + ,

) % . 9

) #

+ ," $ $

" .

" + 2 ; # ,

EXAMPLE

) 2 B

Figure 1.1.21 " "% +"

FACT

() &

&

+

# . ,

Trees as Subgraphs > #/ "

. % 9 .

# /

DEFINITIONS

2 ' " '

" '

) % ' /

' '

) % #

EXAMPLE

) 2 2

" ' #

#!" / ' "


" " " " /

'

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

& " # + ,


#!

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,


5DCC7 K ? K D"

" J"

CCC

54C7 2 4"

" J " CC +2 -" %/="

C@C,

5 C7 L 8 H "

" K

= M " CC

5667 = "

" #" 666

5=67 ="

" -" J/4" 66

+2 -" CC@,


1.2 FAMILIES OF GRAPHS AND DIGRAPHS

!

'/

Section 1.2 Families of Graphs and Digraphs 21

) = 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

() % (* + ,

EXAMPLES

) 2 ) * 6 " . )/

) ; &

+ 6,"

+ ," &

+ ,

) 5=CA7 . / @

Figure 1.2.3 "% +! %" @ '"&


) & &

&

/ :/

. ./ ' / /

:/. ./

) ( / A " ./

Figure 1.2.4 6' +! &&" %" A '"&

Global Symmetry: Vertex-Transitivity( ./ # @

DEFINITIONS

) + *, *

* = # # #

# *

= #

" B

#

) % +B *, #

*

"

.

"

& ) > )

* % B

) + " , )/. &

&

) +

+ , /! /

# +

, 6

)

)

*

&

*

G&


) 6/. / 2 ;

Figure 1.2.5 5"

EXAMPLES

) . G /

' /! &

) % /! J ) /

" #" " "

) J ./"

+ , '

6" "

" "

)

&

FACTS

() 54# 3 7 #

9 & 9 >

. +

() = / # +

"

)

+

*

&

* 6

+

&

1.2.3 Integer-Valued Invariants # / /

# ( #

Cycle Rank $$ 6 "

+ ,


DEFINITION

) * +, # G +

@ ! ! , 8 "

+, " ! # +, +,G +,

EXAMPLE

) ! 6

Figure 1.2.6 " %" ! " 6' '"&

FACTS

() 5 3 7 9

' +" 5DCC" 7,)

' + " ,

'

'

% ' # .

() 5' +, 7 ? #

)

+, &

+, ' ' '

# # # . & .

' " '

+ ,

() ! !

() %

Chromatic Number and k-Partite Graphs' " &"

" & >

/ . ; ;

DEFINITIONS

) % #

+ , " &

+2 " &

, ' , $

" # ,$


) % # .

& '

, $" &

) % ) # ) +

, &

) % ) )/

& > %

' )

"

&

" ," &

EXAMPLES

) - #

) - # #"

# #

) / &

/

Figure 1.2.7 &" +"" &

. *

FACTS

() 5 3 7 % #

+" 5DCC" ;7,

() % )/# )/

() 2 ) " # )/ HJ/

k-Connectivity and k-Edge-Connectivity # 3 $$ *$$

: .

DEFINITIONS

) " -

+,"

# /

) " -

+,

# /


# 01

- " - . -

-

"

) % -

) ) -9"

)/ ) /

) % / -

)

" ) )//

) % ) + ),

)

) % ) + ),

)

Minimum Genus # 3

DEFINITIONS

) + ,

# # # /

+ :,

/

) % 6

1.2.4 Criterion Qualification%

" " # .

DEFINITIONS

) % ! .

+ . ,

) % +

; . ,

) % )/ ) #

+ ;,

) % )/ ) /

. + ,

) % )// ) //

+ ,

) 0+, B

0+, & .

%" # .

0+,


) % . #

+ ,

EXAMPLE

)

Figure 1.2.8 "

FACTS

() 5? 37 9)

5L7 # #

.

5:67 H 2 C #

Figure 1.2.9 !& !

() %

EXAMPLE

)

Figure 1.2.10 "!" %" " ! '"&


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 History of Graph Theory 29

1.3 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 Knigsberg Bridges Problem #. "" 2 " ! /

! # LS# . O

" ! # .

"

dc

c

a

b

C

A

B

D

g

Figure 1.3.1 ' 78


FACTS 5?=CA" 7

() ( @ % :; ? - 0 #/

1 % J#"

" ! #

() ' :@" - " /

#

() - 5-):@7 #

" # 0 # /

1 % :@"

:" # :;

() -T " LS#

# #" #

() - #" #

# 9 %" " " +

, % % & +

# # % # % ," %" "

& & 4 # 9 ."

# LS# # #

() ' #" - # # #

. %" " " # #

/ 0!

1" ./ 9 #

() -T )

' # # "

& #

' # # . " & #

'" " # # "

9 & #

"

/"

() - " # "

#" # " #

% # 43

54)A:7 A:

Diagram-Tracing Puzzles% *$ "%%" 9

# # !

33 # #! P ."

% .


FACTS 5?=CA" 7

() ' A6C ? J 5J)A6C7

#)

" 9

$.# # # "

& 9

J # # "

# '

" 9

() ( / 33 # 5)A7

K ? 5?)A:7 #! $ "

01

() ' AC" ( 9 ! #

# #

&

" # 8 5)A:/7 #

() # LS# # # /

33 3 C ' #

= = 5)AC7

% #

# 2 #

A

B

C

c

d

b

a

b

e

f

g

D

Figure 1.3.2 " 78

Hamiltonian Graphs% # # #

& .

4T $" " + ;,"

& # L!" #

4T " #

FACTS 5?=CA" 7

() % . # / ""

!T #" @ 9 &

# # #!


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


() 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 #

> " # #

"

- ! @

FACTS 5?=CA" 7 5JA:7

() = ! # # ! 0>

1" 5)A;:7 /

() T ! " # #

# + 2 @,

rootroot

Figure 1.3.6 ="" " "

? !

# # #"

G!

G!

G!

G

9

+ ,

+

,

+

,

F 9" # # !

() % A:6" K $0$

$0$

() ' A:" 5)A:7 #

" "

!


() ' AAC" 5)AAC7

# #

4 . * @" #

# 4 JS 5JA7)

# // # #

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 H

H H

CC O

HH

HH

H C C O H

Figure 1.3.7 "" "

FACTS 5?=CA" 7

() T .

" # . +,

# > 2 A

HH

HH

HH

HH

H C C C C

H

H H H H

H

C

C CH

CH

H H

H

Figure 1.3.8 % > !" !"


() 5)A:7 / +!,

# " B

# # ! * A

2

H# ; C A

() = L > # # #

# 9 "

# ' A:A" 5)A::/A7

& # " ! )

- # .# # "

L!I

" /

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

!

"

() ? C6

C6 % ? K L 5?C7 3

" JIT #/ #

1.3.3 Topological Graphs-T 5-):;67

" ' .

' C6" 3 ##

# L33 L ! +AC@/CA6," ! P # #

H #" J " P .

Eulers Polyhedron Formula ! " #

! # # " " 1

)

G 1 *

' : " I " #

-T # 4 "

" # !


FACTS 5?=CA" ;7 5CC7

() "

H# :;6" - # # "

+, & +, # " " "

' # #

# . # # " G * G

() - # ' :; # /

" # # %/8 ? 5?):C7

:C"

() ' A" %/? 5)A7 # -T # /

&

() % " /%/K ? 5?)A7

. "

# 4

-T P "

" / + "

RT , 2 / " ?

G 1 * 6

. " #

"

G 1 *

# & &$" 9

1 $$$ + :,

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

(" ./

." >

3 # -T ! # $

#9 ' " 4 JI ! ?T

AC;/C6 #

() ! JIT ! " /

# 8 J 4 546:7 +, /

)'

*! 4 #

(


K3.35K

Figure 1.3.9 7!"%) &

&

FACTS 5?=CA" A7

() % A6" % 2 8S# 33 /

)

! '

! #

# U

9 ! #

# "

# 4 3)

! &

! # # U

' " # &

() % #" #" "" **

$$ "" # 4 57 " C)

33 " " " =" " -"

" %" " " + 2

6,

# &

W G E

CBA

Figure 1.3.10 +%"+&"&"$

() ' C6 L ! 5L67 #

# &

&

B #

# ( 2! J %

() ' C 4 = 5=7 #

#


4 " "

# # # 8?

() ' C; = 5=;7 3

.

" 5;C7 /C;6

# L !/ + @@,

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)AC7 #

# " 4 T

#

() ' C6" 4 3 567 . 4>T //

# " 8S# # & " /

4 4 # L #" #

# J 2! 527 C" .

' C;" ' L 5L;7 /#

" " @

() 4 /# C; #

# # "

66 8 /

C@6" C@A # D 5D@A7"

= T 5@7 # C@ $ " "

# K ? 5:7 3 #

#&" 3 "

+ :," + = ! 5:7,

() ' 9 CA6 " #

5A;7 " # " 0#

#1 + ::, 4 " " # #

# " 2 /# "

" C:C 4 4 " K J 4!" = 54=:C7

# 6 # # &


1.3.4 Graph Colorings- ! "

" # / # # 2

A;" # +, 01 # % L A:C

' # L % = 4! C:@" #

! L" !>" 4 4" "

#9 # H #" " J "

# 5CC7 8 " #

"

5AA67"


? +? ," 2 + ?," 57 #

) " )"

$$ "

() ' C

" 2! 52

7 # # /

" / ;

#

() % C;6 4 # # /

( . " 4 54@C7 #


() ' 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 #

. # + ,

() &

> C6" ! ="

J

() ' C" ? ! 57 #

. G " 9

+ ;,

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

$$ "

() ' C@"


Factorization% )* )

# #" ./ %

)*&$ )/ #

2 ! # K J

5AC/C67 = 5C/667 + ;,

FACTS 5?=CA" 67

() ' AC" J 5J)AC7 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 + , 6 #

#" #"

+ ," # #

' # !"

" +W ., # #"

!

FACTS 5A7 5??A;7 5?JA@7

() 2 " !

# "


A ' C6" J"

3 %H

3" 2!" 8 K 52K;7

# C ' CA6 # C

# J# 5JA:7 + @,

() $$ "" !

/ " # #! ( !

5@7 # K L! 5L;@7 % "


5?=CA7H ? " - L ?" K ="

,-./0,1./" (.

F J" CCA

57 !>" % #

" ! +C," P@

57 !>" # " ; +C," ;PA

5?@7 !> ? " "

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

5@7 ( !" ( &I #I I

I

" " " & 2

+C@," :P;A

57 ? !" ( !" # # "

: +C," CPC:

5)A7 %/? " 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

5A7 3" # " 3

% +CA," PA

52K;7 3" 2! 8 K" /

/ #" 3 % +C;," CP6

546:7 8 J 4" % " )'

*! +C6:," ;P6

58)A@67 % 8" % "

" # = = " "

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


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

+C;C," @CP:

5;7 % " # " # " +,

+C;," @CPA

57 4 - " J." " @ +K C," 6 +%

C,"

5-@;7 K -" J" $ " : +C@;," CP@:

5-):@7 ? -" +:@, # " 0

" # A +:;," AP6

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"

! " = 4 2 " C:C

54F=:C74 4 " K J 4! =" 6 #

& "

+, : +C:C," P:6

54@7 - 4" 8/ ! $ " "

C +C@," ;;P;;@

5:7" K ? "


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

" :A +C;;," ;P@

54J:7 2 4 - 8 J" )" % J" C:

54)AC67 J K 4 " 8/ " + # +AC6,"

PA

54)AC7 ? 4>"

S

F# J# H##" A +AC,"

::P;A6

54@C7 4 4" F 3


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

6 +A@/," C:PA

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

0

C" H/4" CA@

5?)AA7 - ?" %66 68"


5A;7 H # J " O " "2

,1;< + ' %," ? 8 ? H 6

+CA;," # F J" ;P:

5C:7 H #" " J " " /

"

9 " 4 :6 +CC:," P

5)AC7 = = "

% # ! #

# +

% )," 8"

?" AC

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

# "

+CAA," PC

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

567 4 3" - ! S# J# LS# /

2S" 0$ C +C6," ;;P:C

5@7 = " ( " " +C@," CAP6

5:7= " 3 " "

+C:,"

6:P

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

5:67 = " ( "

C +C:6," ACPC@

5


5=7 4 =" H/# " # & " :" :

+C," ;P:

5=;7 4 =" ( # "

;: +C;," ;6CP;

5=CC7 K =" " : ! + ' 8 K,"

- " CCC

5=67 =" "" J F J" 66


GLOSSARY FOR CHAPTER 1&$&& ) $$

'"1 * +," )

./ / + .

#,

+ , * +,)

./ / + ,

@&&$ "1 P ./

)

. !

!

+ , *

&

6

@&" )

@&" '"&)

&) $

""!") B (

!" P )

"" ) / B #

& +2

" &

# #,

!#!"

) +, . /

&" !&" + "$, P )

+, * +, +, +, * +, +, +, +,

-$$

+ *, P *)

./ "

*

-$$

)

+ *, # 0 1

&" ) $ "

&" '"1 P ) . 9

&"& ! P ) #

/#

&&!" +B *,)

&) # #B $

+& ) ./

&) + ,

01B 01

#

" ) + ,

& +" ,

&") *$"

Chapter 1 Glossary 51

&" "" ) # .

& . '

, $" &

&"

&

) #

" #

&" &

)

& #

&" )+"" &

) )/

& > %

$" " "

&" P ) . #

&&" ) # !

)+&&" ) )

" " #

&&"'"$ ) # ) )/ '

-+, -

+,

&" P ) + , #

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

# + ;,

&"&$ )+&&" ) )

. + ,

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

+ ,

&! ) "$ "

&!) "$

&!"+) #

&!"") $*(

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

&$&)

&$&

) /. "

&$& ) P * +, +, ) # +,

+,G +,

+ $, P . " +,) #

# / +2 "

9 # #,

#!& P ) 9 # .

/

" '"1 * +,)

. " #


" * +,) ./

" # /

" P ) . $$$" !

+ $ ",)

) &

&" "& . " . P )

!

&" + $,) " "

' "

&" ) "

&" P )

"& P # ) ! #

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

) # " * +,

+& P ) /

$

""!") / # # "

$

+&" P ) +

,

./ "

)++&&" ) )

#

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

' -

+, -

+,

+""' )

"$ &

)

") "

! ) ! .

+ .

,

! " P )

&" P ) #

)+&" P ) )/ #

")

" P )

B

) # / /

. .


!)

* +,) $ "

" "

" ) +

; . ,

) $

$&! +

) /! / #

$&! ) / +

, 6

,

&&) #

P . ) #

!& ! P *

) #

./

'" P "

" '"1 P ) . 6

& ) # H/

)

*

'" P )

B " #

) #&

)

)

"

& #&

&

+, +,

) #& )

"

+, +,

@

+ ",) G

+G, * +, +,

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

@

) G 0&1

@

) 0&1

" %)) #

P ) 0+, ./ /

" &

.

) % # " .

0+,

) &*"

) &*"

! ! + !, ) #

# # /

+ :, /

H) 2

+, 2+,


1 ) "

$ "

!"+&)

!"+) "

!") / ." # /

P .) & .

" P ' ) '

! ) ./ / B &

&"

) / / &

" '"1 P )

+, *

" P )

+, *

!" P . ) #

"$ &" ) #

3+"" ) ./ 3 #

+ " ," &

#

"" ") 3*"

") .

" &) $ .

" ) " .

" (

) /. "

+N " # (

# ,

5" ) 6/. / " ;/

" / &

) 6" " #

"& ) !

"& ) / O " #"

" "

!&") $ "$

) &

+&) / &

'"1+&) ./ &

! P ) $$$" !

! ) . ' )*

. )


+) &

+!. + P ) & % *

%

) / /

) / /

1) . / -

' )*"( / * ) G

)" + *, )/. &)

&

! P ) # .

" P ) #

" A'"1B P ) # # #

# 6 # #

"$ &&" ) .

. !

! )

"

!) -

") $

") !

")

" P ' ) '

"' &

) .

"' %). ". ") ! 3

!$ P . #)

# +" /,

! " P ) ./

/ & " " ./ /

'&)

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

'"1+& P ) ./

$

'"1) # " * +,

'"1+""' )

%) P ) 9 *

"

* "

" &) ! . .

" ) ! . > .

%)$ &&" ) "

%" P ) + $, #

Chapter 2GRAPH REPRESENTATION

2.1 COMPUTER REPRESENTATIONS OF GRAPHS

2.2 THE GRAPH ISOMORPHISM PROBLEM

2.3 THE RECONSTRUCTION PROBLEM

2.4 RECURSIVELY CONSTRUCTED GRAPHS ! !

"

" "

GLOSSARY

Section 2.1 Computer Representations of Graphs 57

2.1 COMPUTER REPRESENTATIONS OF GRAPHS

Introduction

! "

# #

$ %

#

2.1.1 The Basic Representations for Graphs # & '

&

DEFINITIONS

( ) *+ !,

- * +

( ) *+ !,

-

( . ) *+, ' ' * +

&

( . ) *+, ' '

&

( / *

+, *

+,,

*

+ '

'

( ) *+ #

( ) *+ #

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 +


REMARKS

%( 4 , 06781, 0

6791, 0:; ) *+ , & ' &

/ *

+ 6#, # & ', #

# ' , # # & #

* + 4 , & '

#

%( ? & ,

# & ( ' '

2.1.2 Graph Traversal Algorithms@ %

A! !

/ / 4

/ , & #% #

Depth-First Search

ALGORITHMA! .,

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

# ' , % ,

& # B! C

# *+ % #

DEFINITIONSA , !

%

, # (

( * + # ! '

* +

( * + '

!

( * + '

!

(


Algorithm 2.1.1: &$# '

# ( ) *+, # ) 01

& '

$# ( !

!( *+

! () ! !

01 () 2

! () ! !

01 ) 2

*+2

!( *+

01 () 2

! ' 01 !

01 )

*+2

FACTS

$( A! % * 5 + ) *+

$( . # ! ' # B*C

B+C, / !

' #

$( . ! ,

%

REMARKS

%( A!

=D3 08 681 !

%( A!

!

Breadth-First Search! / ' .

! ' #

. !

' ,


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 + ) *+


$( # * +

01

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> '

# ) *+ *

+ *

+

REMARKS

%( 4 4>

06781 0:;


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 ) *

0 1 5

0 1+ .

' # 4> FE

Transitive Closure. # & # %# # ' '

' ) *+ >

) *+ #

& ' , # # ' $ $ 0 1

, 3 # > $

& '

# 4> '

,

FACT

$( $%!*+

*

+ *

+

REMARKS

%( 0>G1


Algorithm 2.1.4: ,#- ."!#(

# ( ) *+, # ) & '

0 1

$# ( ' $ 0 1 # $ 0 1

, 3 #

!( $%!*+

! () ! !

! () ! !

$ 0 1 () 0 12

! () ! !

! () ! !

! () ! !

0 1 ) "#

0 1 () 0 1!0 12

%( ; $

0 1 ) ' '

' , '

>

%

0 1 ) %

0 1 ! %

0 1 %

0 1

# ! . ' #

# FE

2.1.4 Applications to Pattern Matching & !

%

"

' '

#


( ?4 * '

! ' # *

(

?4 !

( . + , +

*&* &

( ; A!

) )

)

,

, !

( ; H ! H

! #(

, '

) ' )

4 H, '

. ' + ,

* 5 + ' + ,

' + ,

*

+ ' +

> # '

F

5,

5 4 ', ***-

++ 5 +

# -

5 ' -

3

Kleenes Algorithm


Algorithm 2.1.5: /"0# "!

# ( ) *+, # ) , '

0 1

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

!( .

*+

! () ! !

! () ! !

%

0 1 () 0 12

! () ! !

%

0 1 () )5 %

0 12

! () ! !

! () ! !

! () ! !

%

0 1 () %

0 1 5 %

0 1 *%

0 1+

%

0 12

! () ! !

! () ! !

%0 1 () %

0 12

' ,

%

0 1 *%

0 1+

%

0 1+ #

( , " , #

'

%( 4> FE #

%

0 1 () *%

0 1 %

0 1 5 %

0 1 . 4

> # E #

, 4>

*+

%( FE #

%

0 1 () %

0 1 5%

0 1 %

0 1 # 5

!

%( , 6

, 7 FE

06781

%( @ %

! ?4 ' ! '

%( 4 ! '

0=3,


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


2.2 THE GRAPH ISOMORPHISM PROBLEM

I

! /

.

D :'

Introduction , !,

# # . ' '$,

' . ' *%+

*'*

++, *'* ++

'$ *'*

+ '$

# '

, , !

#

2.2.1 Variations of the Problem

DEFINITIONS

( #

) *

+

) *

+ ,

, , , (

, #

* &

, *&

,**+,*&+

< & & ,

( # L ,

# , ! . L

, $

, # #

( #

) *

+

) *

+

, (

,

,

,*+

( *+, #

, #

) *

+

) *

+ ,

,

Section 2.2 The Graph Isomorphism Problem 69

, , (

,

, ,*+

2

,

, ) ,*+

( / #

, /*

+ ) /*

+

(

/

*+ )

, # ) *

+ 0 )

*!

!

+ &

!

( )

'

FACTS

$( #

# &

' 1 ,

) 1

1

# ! '

# & L

$( /

'$

$( /

'$

EXAMPLES

( -' # 2

,

%2 , %


0:91, 0AA;881, 04931, 04M

#

FACTS

$( 08G1 /

$( 0A431 / # /

/ # #

' , /

/

$( 0AI=G1 . # #

,

Reconstruction from the Characteristic Polynomial Deck # #

! , # ?

> ; , % #

/ ' '

0=, ; ; F, @ ,

*=9+, 3J

F9D1 > ; F, @

Section 2.3 The Reconstruction Problem 97

0=1 I %, , # #

$# = *==+, 9J8

0=D1 I %, %, %# # # ;

# 3 *==D+, GJ8

0=91 I %, ,

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

0=1 , ,


0

Section 2.4 Recursively Constructed Graphs 99

2.4 RECURSIVELY CONSTRUCTED GRAPHS ! !

"

" "


, * + A!

6#, !

* +

EXAMPLE

( 4

Figure 2.4.1 %(#- !#(! !

Series-Parallel Graphs4 ,

.

0A GD1 4 2

$ ! # , ,

#

Figure 2.4.2 2!#&""" ##&""" #

4#, # !

DEFINITION( # * +

! #(

" *

+ * + #

)

)

" *

+#

*

+

#

#

" *

+ #

*

+

#

#

#

" *

+ #

*

+

#

2 #

( &%% ! # #

, , !

EXAMPLE

( ! 4

! 2 #


Figure 2.4.3 .!!#! !!# ! ##&""" #

-Trees and Partial -Trees

DEFINITIONS

( ' ,.

, # 5 * +

' &

.

,

(

. , .

( * +

( /

" ' /

FACTS

$( ,

$(


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


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 )*& *& #


Treewidth- Graphs #%


Pathwidth- Graphs

DEFINITIONS

( #

/ '

,

, !

( '

( # %

( #

EXAMPLE

( # 4 3 '

!

#

$

Figure 2.4.10 &!!#!

Branchwidth- Graphs

DEFINITIONS

( ) *+ *$ +, # $

# ' ' &

$

( . ' $ , *$ +

( ; *$ + ) *+

$ '

$

$ $ , # *

+ *

+

( *$ + '

$


( # %

( #

FACTS

$( 0


Figure 2.4.12 %(#- !#(! ! &"

REMARKS

%( , # * $+ #

) $ ,

%*++ %*#++, # #

*, .

+

%( '

,

# 4 ,

) *

$

+,

& ' ; ) * $+ #

(

"

(

*

+ ) *

+$

)

" * + (

*+ $ $ $

" (

)

*

+ ) *

+$

$

$

" (

*%+*

+& *

+ *

+

. ' ,

* + # *

+ )

CographsDEFINITION

( ! #(

" # '

" .

, &

" .

,

, #

%

*

+ #

:


EXAMPLE

( 4

! *+ #

Figure 2.4.13 .! !#(!

FACTS

$( 0:; 91

$( 0:; 91

Cliquewidth- Graphs .+ # 0:-=1

%

DEFINITION

( ; 0%1 #

! #(

" # *+ ) *+ 01 / #%

" .

/ # 01,

*+ &

/ #

*+ *

+

/ # , # *

+

*

+ *

+ ) *

+ )

*+ *

+

/ # , # *

+

# #

REMARK

%( A! 9 ! / # / #

/ # / #

,

, ' ,

, ,

. , .

. %, / ! (

# &


- # , # #

-K ) $),(

3.3.1 Basic Definitions and Examples

" " #

/ 0

DEFINITIONS

2 ! ! -

# " " / 0

2 "

#

2 " / 0 " #"

/ 0 + / 0

Section 3.3 Tournaments 157

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

/

0

@

-

!

! "

"

2 $C9( /0 - /

0 - "

"" "

"!

/0

/

0

S

/0

/

0

S

5

A # " /0 "

9 C ' : H & ,

/0 9 C' 9C' 'HH, &C' &:,C'

EXAMPLES

2 9

158 Chapter 3 DIRECTED GRAPHS

Figure 3.3.1 "

2 )


REMARK

2 4! #

/0 6 )

!

5 - ?

EXAMPLES

2 &- #

" "

9! "! #

/0! #

5 )

5 9 ' H

1

2

3

45

6

7

8

0

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

" -


2 $:( 3 # - % A # %

!

4 " %-

REMARK

2 $HH( #

"- B A

- #

" B#"!

/ $H9(0

CONJECTURE=

=* 2+ $

T

'9(: 7" -"

# "

3.3.2 Paths, Cycles, and ConnectivityI - " #

" "

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

A #

DEFINITIONS

2 / 0 & -

" & / 0 &

" & /B

9C0

2 & / 0 "

" &!

EXAMPLE

2 #

! #/ 9 ' H0 "

, ' C 9 H : ,

" " !

REMARK

2 :

4 A " "

FACTS

2 $T9( 7" "! " -


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 $B8 'C(

"

2

# A" 4" - $'H(

/0 "

/0

/0 4

/0 4 /, 0

/0 "

! /! $( 0


2 7" /

0- " -

Cycles and Paths in Tournaments

FACTS

2 $':( 7"

-! !

-!

- / $H,( 0

2 $K:( 7"

-! H!

-! 9 - / $H,( 0

2 $:9( " / 0

- ! # :!

" -! - ! - /

$H,( 0

2 $H,( " / 0

- ! # ,!

" - - ! -

/ $)G&:(0

Hamiltonian Cycles and Kellys ConjectureCONJECTURE>&&)* + / $'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

2 $BU&( " "! "

!

- " -1

2 $HC( 7

-


Higher Connectivity

DEFINITION

2 & % / % 0 " % #

" &! &

FACTS

2 $H,( 7" -

"! A - "

" 9-

"! A

-

2 $H:( 3 %- 3 % #

! - 3

2 $H9( " % !

!

/%0- !

2 $&( -! - ! " - - ! '!

# "-1 - -!

4 # / 90 /

- 5 ! # # $HC( $),,(0

2 $ )%,( 3 %- - # H%! %

"-1

"

Anti-Directed Paths

, ! "

3

!

/

0!

3 ! %

%

DEFINITION

2 / 0 & 4

&

&

EXAMPLE

2 # - -

9


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 "

$H'(0

3.3.3 Scores and Score Sequences ,! B ) % $%C(!

8


# - ! 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(


The Second Neighborhood of a Vertex

DEFINITION

2 % " & !

(

/0!

" & - -

! (

/0 5 $

(

/0 ((

/0

FACTS

2 $&'( 7" "

(

/0

(

/0

2 $B,,( 3 !

# " '

CONJECTURE7)* ' + / $&'(02 7" & -

" $ #

(

/$0

(

/$0

3.3.4 Transitivity, Feedback Sets, Consistent Arcs3 -

/ - 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

" #


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 ! " "

"

? "

*/0

? "

"! "

#

/$'&! :(02

( 9 C ' : H & ,

*/0 9 ' & , 9 , C 99-9'

2 $I:,! 8&9! T&9( 3 /0

" --

" - # /0 "!

/0 5

9 :

9 H

C 9 :

' H

/'':0 /0

? C9

/0

/'CC0? : CC

2 $B&H( $))+&H( 4 5 /

0! #

!

" "

-

" " "! !

#" / 0 / $,( !

$B&( -

0


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 ! / 0 "/ 0

2 $N:'( ,! /0 ! "/0 ! ,!

-

6

-1 / $:C(

$),( $3&C(0

3.3.5 Kings, Oriented Trees, and ReachabilityN => "

# # =6>

/ #6 -

$%C(0 $H,(

7 # " -

"" = ->

6 =" "> "

/ H #0 "! 6

< !

- /!

$&'(0

DEFINITIONS

2 " " " !

- -

2 " " " !

- -

2 %! % ! " " 6 %

" 6

FACTS

2 $GC( 7" 6 3 ! " "

6 $%C(

2 $H,( " % ! - #

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


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 - #

"

CONJECTURE7* + / $+H(02 7" / 0- " -

"

EXAMPLE

2

H 9 # C

'-

Figure 3.3.5 "

REMARK

2 8

#

1! 4 =->

4

E

/ 0- " ."


, "

/

$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

CONJECTURE0 ? + / $+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 # / $%&'( $,,(

" 40

FACTS

2 $+H( 3 # # !

" " " 5 !

/ $H9( 0

2 $HH( 3 #

- " - #

! " " " 5 !


3.3.6 Domination3 "

"

B#"!

!

# 5/ 0 5 % " %

6#! # "! " 6#

" % /0 &

DEFINITIONS

2 " "

" "

2

"- ! 1 #"

2 /0 # #

" "!

2

! 5/ 0

EXAMPLE

2 % " - # "- #

#" " / 0 -

# " " " -! " /0

" - / " 0 ! 6 !

" " !

5/ 0 5

FACTS

2 $):( " %!

%

! # / 90 4

- % / $BB,(0

2 $%&H( 6

# #

"!

3

!

- ! 6

# #

"

2 $%&&(

6 ! !

# "

" 1

" #


3.3.7 Tournament Matrices #6 )

K + 8 K I

/ E $):(0E

#6 B K $'9( # " #

! ! F " E

#6 6 $'C( #

#


FACTS

2 $)'H( %

"

! #

, !

/ 0'!

/ 0'

! 5 "! 6 -

2 " 6! !

2

6

2 $ )NI&(

!

6 "E

" B /!6

6 5 20

# "

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 $&(

-

" -

" / -

"0 " " " => "

"

=#> "

" 7 -!

" ! 4 #

2 "

/

0! " " 1

! " # 6" ! #

" => " /

0! " "

" =">

! "


" - / 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-

!

!

3 #

!

!

!

## " " #

! ! " " "

! " " ! ! "

1

! #

1 " ! 1 "! ! 1 "

Figure 3.3.7 " +)


Tournaments That Are Majority DigraphsFACTS

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 " #!

" /

0 "! "

A "! #

"! #

"!

2 )" 1 - /

0 "

" " ! " ""

" /! /0

"0 " 1 "

3 !

! !

" 6 /

0

EXAMPLE

2 )" / " 0 1 #

H! "

" #


Figure 3.3.8 -+)

FACT

2 $::( ! "

"

/ ! $&(0

Division Trees and Sophisticated Decisions

DEFINITIONS

2 )" /

0! /

0

!

! !

"

- -

4 /

0E

/

0E !

, ! " " #

4 /

0!

/

0! # " " ? !

/

0!

/

0

2 % 1 - /

0 -

" " "

"

" /

0 ! #

" " " 1

# # " 6

" " E "! , ! !

" " " 1 #

# " " ?

FACTS

2 $HC( " -

"

" 4

" " -

2 8 " "

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

2 $&:( # "

" -

- !

!

EXAMPLES

2 " / $0 # & )"

1 #! " , "


"

!

" 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 - /

0

" 6 " 3" A

4 $

$

$

#2 $

5

! ! !

$

5

2/$

0

$

#

$

References

$

T

'9(

T

T! I! MH ' " ! !

$!


$':( !

! ! %! /! ,

/&':0! HMHC

$:H( N ! 4 !

&! /&:H0! 9M9&

$:9( ! N I

! !

&! ! / : /&:90! MH

$H( ! 9- !

%! /&H0! M&

$B&H( B! " #

! ! 9& /&&H0! HCM&

$"H,( I "! 4 ! &! 9 /&H,0!

C:M'9

$HC( K 6! " ! $

0' /&HC0! &CM,'

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

! &! ! / C /&&0! M

$B&( N ! % + 6 B! #

! G" 3 &! 1 /&&0! HM&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 TT! . 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


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

" F ! )! ! ! )! (! '4 4 , /&:'0! HM&

$H( K- ! 2 "! &!

C /&H0 M9

$+:C( % + 6 K +! " !

M9H ) ! ! $ ! 1567! !

&:C

$)'H( 3 )! . !

/! ! %! $! :9 /&'H0! MC

$):( 3 )! 6 ! 3

! C /&:0! MH

$%H( V %! J ! &! !

/ /&H0! 'M::

$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( % "!

! /! %! / ! ' /&C90!

M9,

$7'9( I 7U % ! .

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


$H( + <

G! .

! &! ! / /&H0! HM

$&'( ! 4 2 F 1! &!

/&&'0! 9M9H

$%&H( ! K %! < N !

! &! & /&&H0! ,M,

$%&&( ! K %! ! &! ! %! ! ! /&&&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( )! )TT! T I +! ---

" ! &! ! / : /&&H0! HM&'

$)G&:( * ) % G6! ! ! %! :&

/&&:0! :MC

$B,,( B"! . B ! &! ! /

/,,,0! M

$BU&( BU6"! B ! ! ! !

/&&0! CM

$B''( B % ! ! !

%! % : /&''0! M9'

$B8 'C( B! R 8! #! $ %

' ! K + O ! &'C


$B,,( B" T! . B 2

F 1! &! ! / :H /,,,0! 9M:

$B,,( B" T! 2

F 1! &! C /,,,0!

99MC'

$BU&( BU6" ! !

/&&0! M,

$BU&:( BU6" ! . B

! &MC " " 8" 1559:!

J" I! &&:

$3&C( ) 3

6! 6 ! 7 &! ! /&&C0! &

$K:( . K6! ! ! J" !

&:

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

! ! /&&H0! M

$%C( B ) %! .

333

! /! %! / ! C /&C0! 9M9H

$%&:( 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


$'( K + % !

! !

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

$8&9( G 8 %! I6 !

! , /&&90! 'M''

$I:,( 7 I6 N ! 1 7U

! &! ! & /&:,0! CMH

$I,( G I"!

# ! ! /,,0! C&M

C&'

$IH9( G I"! B ! C&M'&

" $" 15' /&: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! %! &:&


$:( N 7 #!

4" 6#

B ! &! ! /&:0! MH

$H9( N

! 6 ! :MH

! I! &H9

$BB,( N ! ! B B!

! ! ,,

$+H( N 8 +! 7 - !

$ $! %! H /&H0! ::MH:

$:9( ! - B ! &! !

/ ' /&:90! 9M9

$'9( B K !

handbook of graph theory

Documents

handbook of discrete

fundamental number theory

handbook of graph theorydarrel

information storage

publishnd information

pm tains information

n d b o o

combinatorial algorithms