Graph Theory and Its Applications, Second - Jonathan L. Gross.pdf
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DISCRETE MATHEMATICS AND ITS APPLICATIONS Series Editor KENNETH H. ROSEN
Graph Theory and Its Applications Second Edition
DISCRETE MATHEMATICS and ITS APPLICATIONS Series Editor
Kenneth H. Rosen, Ph.D. Juergen Bierbrauer, Introduction to Coding Theory Kun-Mao Chao and Bang Ye Wu, Spanning Trees and Optimization Problems Charalambos A. Charalambides, Enumerative Combinatorics Henri Cohen, Gerhard Frey, et al., Handbook of Elliptic and Hyperelliptic Curve Cryptography Charles J. Colbourn and Jeffrey H. Dinitz, The CRC Handbook of Combinatorial Designs 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, Second Edition Jonathan L. Gross and Jay Yellen, Graph Theory and Its Applications, Second Edition 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, Second Edition Daryl D. Harms, Miroslav Kraetzl, Charles J. Colbourn, and John S. Devitt, Network Reliability: Experiments with a Symbolic Algebra Environment Derek F. Holt with Bettina Eick and Eamonn A. O’Brien, Handbook of Computational Group Theory 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 William Kocay and Donald L. Kreher, Graphs, Algorithms, and Optimization 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
Continued Titles Richard A. Mollin, Algebraic Number Theory Richard A. Mollin, Codes: The Guide to Secrecy from Ancient to Modern Times Richard A. Mollin, Fundamental Number Theory with Applications Richard A. Mollin, An Introduction to Cryptography Richard A. Mollin, Quadratics 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 Jörn Steuding, Diophantine Analysis 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
DISCRETE MATHEMATICS AND ITS APPLICATIONS Series Editor KENNETH H. ROSEN
Graph Theory and Its Applications Second Edition
Jonathan L. Gross Jay Yellen
Boca Raton London New York
Copyright Jonathan L. Gross and Jay Yellen
CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2006 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20110713 International Standard Book Number-13: 978-1-4200-5714-0 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com
PREFACE
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ABOUT THE AUTHORS Jonathan Gross is Professor of Computer Science at Columbia University. His research in topology, graph theory, and cultural sociometry has earned him an Alfred P. Sloan Fellowship, an IBM Postdoctoral Fellowship, and various research grants from the Office of Naval Research, the National Science Foundation, and the Russell Sage Foundation. Professor Gross has created and delivered numerous softwaredevelopment short courses for Bell Laboratories and for IBM. These include mathematical methods for performance evaluation at the advanced level and for developing reusable software at a basic level. He has received several awards for outstanding teaching at Columbia University, including the career Great Teacher Award from the Society of Columbia Graduates. His peak semester enrollment in his graph theory course at Columbia was 101 students. His previous books include Topological Graph Theory, coauthored with Thomas W. Tucker. Another previous book, Measuring Culture, coauthored with Steve Rayner, constructs network-theoretic tools for measuring sociological phenomena. Prior to Columbia University, Professor Gross was in the Mathematics Department at Princeton University. His undergraduate work was at M.I.T., and he wrote his Ph.D. thesis on 3-dimensional topology at Dartmouth College.
Jay Yellen is Professor of Mathematics at Rollins College. He received his B.S. and M.S. in Mathematics at Polytechnic University of New York and did his doctoral work in finite group theory at Colorado State University. Dr. Yellen has had regular faculty appointments at Allegheny College, State University of New York at Fredonia, and Florida Institute of Technology, where he was Chair of Operations Research from 1995 to 1999. He has had visiting appointments at Emory University, Georgia Institute of Technology, and Columbia University. In addition to the Handbook of Graph Theory, which he coedited with Professor Gross, Professor Yellen has written manuscripts used at IBM for two courses in discrete mathematics within the Principles of Computer Science Series and has contributed two sections to the Handbook of Discrete and Combinatorial Mathematics. He also has designed and conducted several summer workshops on creative problem solving for secondary-school mathematics teachers, which were funded by the National Science Foundation and New York State. He has been a recipient of a Student’s Choice Professor Award at Rollins College. Dr. Yellen has published research articles in character theory of finite groups, graph theory, power-system scheduling, and timetabling. His current research interests include graph theory, discrete optimization, and graph algorithms for software testing and course timetabling.
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CONTENTS Preface 1.
INTRODUCTION to GRAPH MODELS (( ? # (# % @ (: (1 4 $ ## (3 * ? #! (: ; % 1. () F $9 4 $ 3! (+ ' :( :1
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297
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40
Chapter 1 INTRODUCTION TO GRAPH MODELS
$ " $ ¼ " "
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41
Paths, Cycles, and Trees
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42
Chapter 1 INTRODUCTION TO GRAPH MODELS
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43
Paths, Cycles, and Trees
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Chapter 1 INTRODUCTION TO GRAPH MODELS
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Paths, Cycles, and Trees
45
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Chapter 1 INTRODUCTION TO GRAPH MODELS
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47
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1.6 VERTEX AND EDGE ATTRIBUTES: MORE APPLICATIONS
# & " 2*
Four Classical Edge-Weight Problems in Combinatorial Optimization
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49
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Chapter 1 INTRODUCTION TO GRAPH MODELS
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53
GLOSSARY
' 0 *' " " 0 ' 0 1 ' ( '
! ' *
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& . $ . + , . . 2 7 + , . . 2 7 * '
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Chapter 1 INTRODUCTION TO GRAPH MODELS
2 ' * * * (
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55
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Chapter 1 INTRODUCTION TO GRAPH MODELS
+ ' * 2C " Q )
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2
STRUCTURE AND REPRESENTATION 2.1 Graph Isomorphism 2.2 Automorphisms and Symmetry 2.3 Subgraphs 2.4 Some Graph Operations 2.5 Tests for Non-Isomorphism 2.6 Matrix Representations 2.7 More Graph Operations
INTRODUCTION
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STRUCTURE AND REPRESENTATION
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GRAPH ISOMORPHISM
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59
Graph Isomorphism
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STRUCTURE AND REPRESENTATION
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Graph Isomorphism
61
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STRUCTURE AND REPRESENTATION
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BINARY-TREE TRAVERSALS
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135
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Chapter 3 TREES
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Binary-Search Trees
137
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147
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153
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4 SPANNING TREES
4.1 Tree Growing 4.2 Depth-First and Breadth-First Search 4.3 Minimum Spanning Trees and Shortest Paths 4.4 Applications of Depth-First Search 4.5 Cycles, Edge-Cuts, and Spanning Trees 4.6 Graphs and Vector Spaces 4.7 Matroids and the Greedy Algorithm
INTRODUCTION
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164
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TREE-GROWING
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165
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SPANNING TREES
Discovery Order of the Vertices
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169
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189
Applications of Depth-First Search
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Chapter 6 OPTIMAL GRAPH TRAVERSALS
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Gray Codes and Traveling Salesman Problems
GRAY CODES AND TRAVELING SALESMAN PROBLEMS
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9.3
393
Edge-Colorings
EDGE-COLORINGS
( # #5 # 75 * 75 #5 #5 * #5 75 #5 * The Minimization Problem for Edge-Colorings
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Chapter 9 GRAPH COLORINGS
Tutte’s 1-Factor Theorem
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Factorization
409
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411
Supplementary Exercises
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SUPPLEMENTARY EXERCISES
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412
Chapter 9 GRAPH COLORINGS
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413
Glossary
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425
Domination in Graphs
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504
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SPECIAL DIGRAPH MODELS
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12.2 DIGRAPHS AS MODELS FOR RELATIONS
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NETWORK FLOWS AND APPLICATIONS
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NONPLANAR LAYOUTS
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APPENDIX
A.1 Logic Fundamentals A.2 Relations and Functions A.3 Some Basic Combinatorics A.4 Algebraic Structures A.5 Algorithmic Complexity A.6 Supplementary Reading
A.1
LOGIC FUNDAMENTALS
Propositional Logic
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Types of Proof
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SOME BASIC COMBINATORICS
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