Graph Theory with Algorithms and its Applications

Santanu Saha Ray

Graph Theorywith Algorithmsand its Applications

In Applied Science and Technology

123

Santanu Saha RayDepartment of MathematicsNational Institute of TechnologyRourkela, OrissaIndia

ISBN 978-81-322-0749-8 ISBN 978-81-322-0750-4 (eBook)DOI 10.1007/978-81-322-0750-4Springer New Delhi Heidelberg New York Dordrecht London

Library of Congress Control Number: 2012943969

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This work is dedicated to my grandfather lateSri Chandra Kumar Saha Ray, my father lateSri Santosh Kumar Saha Ray, my belovedwife Lopamudra and my son Sayantan

Preface

Graph Theory has become an important discipline in its own right because of itsapplications to Computer Science, Communication Networks, and Combinatorialoptimization through the design of efficient algorithms. It has seen increasinginteractions with other areas of Mathematics. Although this book can ably serve asa reference for many of the most important topics in Graph Theory, it evenprecisely fulfills the promise of being an effective textbook. The main attention liesto serve the students of Computer Science, Applied Mathematics, and OperationsResearch ensuring fulfillment of their necessity for Algorithms. In the selectionand presentation of material, it has been attempted to accommodate elementaryconcepts on essential basis so as to offer guidance to those new to the field.Moreover, due to its emphasis on both proofs of theorems and applications, thesubject should be absorbed followed by gaining an impression of the depth andmethods of the subject. This book is a comprehensive text on Graph Theory andthe subject matter is presented in an organized and systematic manner. This bookhas been balanced between theories and applications. This book has been orga-nized in such a way that topics appear in perfect order, so that it is comfortable forstudents to understand the subject thoroughly. The theories have been described insimple and clear Mathematical language. This book is complete in all respects. Itwill give a perfect beginning to the topic, perfect understanding of the subject, andproper presentation of the solutions. The underlying characteristics of this book arethat the concepts have been presented in simple terms and the solution procedureshave been explained in details.

This book has 10 chapters. Each chapter consists of compact but thoroughfundamental discussion of the theories, principles, and methods followed byapplications through illustrative examples.

All the theories and algorithms presented in this book are illustrated bynumerous worked out examples. This book draws a balance between theory andapplication.

Chapter 1 presents an Introduction to Graphs. Chapter 1 describes essential andelementary definitions on isomorphism, complete graphs, bipartite graphs, andregular graphs.

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Chapter 2 introduces different types of subgraphs and supergraphs. This chapterincludes operations on graphs. Chapter 2 also presents fundamental definitions ofwalks, trails, paths, cycles, and connected or disconnected graphs. Some essentialtheorems are discussed in this chapter.

Chapter 3 contains detailed discussion on Euler and Hamiltonian graphs. Manyimportant theorems concerning these two graphs have been presented in thischapter. It also includes elementary ideas about complement and self-comple-mentary graphs.

Chapter 4 deals with trees, binary trees, and spanning trees. This chapterexplores thorough discussion of the Fundamental Circuits and Fundamental CutSets.

Chapter 5 involves in presenting various important algorithms which are usefulin mathematics and computer science. Many are particularly interested on goodalgorithms for shortest path problems and minimal spanning trees. To get rid oflack of good algorithms, the emphasis is laid on detailed description of algorithmswith its applications through examples which yield the biggest chapter in thisbook.

The mathematical prerequisite for Chapter 6 involves a first grounding in linearalgebra is assumed. The matrices incidence, adjacency, and circuit have manyapplications in applied science and engineering.

Chapter 7 is particularly important for the discussion of cut set, cut vertices, andconnectivity of graphs.

Chapter 8 describes the coloring of graphs and the related theorems.Chapter 9 focuses specially to emphasize the ideas of planar graphs and the

concerned theorems. The most important feature of this chapter includes the proofof Kuratowski’s theorem by Thomassen’s approach. This chapter also includes thedetailed discussion of coloring of planar graphs. The Heawood’s Five color the-orem as well as in particular Four color theorem are very much essential for theconcept of map coloring which are included in this chapter elegantly.

Finally, Chapter 10 contains fundamental definitions and theorems on networksflows. This chapter explores in depth the Ford–Fulkerson algorithms with neces-sary modification by Edmonds–Karp and also presents the application of maximalflows which includes Maximum Bipartite Matching.

Bibliography provided at the end of this book serves as helpful sources forfurther study and research by interested readers.

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Acknowledgments

I take this opportunity to express my sincere gratitude to Dr. R. K. Bera, formerProfessor and Head, Department of Science, National Institute of TechnicalTeacher’s Training and Research, Kolkata and Dr. K. S. Chaudhuri, Professor,Department of Mathematics, Jadavpur University, for their encouragement in thepreparation of this book. I acknowledge with thanks the valuable suggestionrendered by Scientist Shantanu Das, Senior Scientist B. B. Biswas, Head ReactorControl Division, Bhaba Atomic Research Centre, Mumbai and my formercolleague Dr. Subir Das, Department of Mathematics, Institute of Technology,Banaras Hindu University. This is not out of place to acknowledge the effort of myPh.D. Scholar student and M.Sc. students for their help to write this book.

I, also, express my sincere gratitude to the Director of National Institute ofTechnology, Rourkela for his kind cooperation in this regard. I received consid-erable assistance from my colleagues in the Department of Mathematics, NationalInstitute of Technology, Rourkela.

I wish to express my sincere thanks to several people involved in the prepa-ration of this book.

Moreover, I am especially grateful to the Springer Publishing Company fortheir cooperation in all aspects of the production of this book.

Last, but not the least, special mention should be made of my parents and mybeloved wife, Lopamudra for their patience, unequivocal support, and encour-agement throughout the period of my work.

I look forward to receive comments and suggestions on the work from students,teachers, and researchers.

Santanu Saha Ray

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Contents

1 Introduction to Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Definitions of Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Some Applications of Graphs . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Incidence and Degree. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Complete Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.6 Bipartite Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.6.1 Complete Bipartite Graph . . . . . . . . . . . . . . . . . . . . . . 71.7 Directed Graph or Digraph . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Subgraphs, Paths and Connected Graphs . . . . . . . . . . . . . . . . . . . 112.1 Subgraphs and Spanning Subgraphs (Supergraphs) . . . . . . . . . . 112.2 Operations on Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Walks, Trails and Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 Connected Graphs, Disconnected Graphs, and Components . . . . 152.5 Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3 Euler Graphs and Hamiltonian Graphs . . . . . . . . . . . . . . . . . . . . 253.1 Euler Tour and Euler Graph . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 Hamiltonian Path. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2.1 Maximal Non-Hamiltonian Graph . . . . . . . . . . . . . . . . . 273.3 Complement and Self-Complementary Graph . . . . . . . . . . . . . . 31Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4 Trees and Fundamental Circuits. . . . . . . . . . . . . . . . . . . . . . . . . . 354.1 Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.2 Some Properties of Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

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4.3 Spanning Tree and Co-Tree . . . . . . . . . . . . . . . . . . . . . . . . . . 404.3.1 Some Theorems on Spanning Tree . . . . . . . . . . . . . . . . 40

4.4 Fundamental Circuits and Fundamental Cut Sets . . . . . . . . . . . . 414.4.1 Fundamental Circuits. . . . . . . . . . . . . . . . . . . . . . . . . . 414.4.2 Fundamental Cut Set . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5 Algorithms on Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.1 Shortest Path Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.1.1 Dijkstra’s Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 495.1.2 Floyd–Warshall’s Algorithm. . . . . . . . . . . . . . . . . . . . . 57

5.2 Minimum Spanning Tree Problem . . . . . . . . . . . . . . . . . . . . . . 665.2.1 Objective of Minimum Spanning Tree Problem . . . . . . . 675.2.2 Minimum Spanning Tree . . . . . . . . . . . . . . . . . . . . . . . 68

5.3 Breadth First Search Algorithm to Find the Shortest Path. . . . . . 785.3.1 BFS Algorithm for Construction of a Spanning Tree. . . . 79

5.4 Depth First Search Algorithm for Constructionof a Spanning Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6 Matrix Representation on Graphs . . . . . . . . . . . . . . . . . . . . . . . . 956.1 Vector Space Associated with a Graph. . . . . . . . . . . . . . . . . . . 956.2 Matrix Representation of Graphs . . . . . . . . . . . . . . . . . . . . . . . 96

6.2.1 Incidence Matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 966.2.2 Adjacency Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016.2.3 Circuit Matrix/Cycle Matrix . . . . . . . . . . . . . . . . . . . . . 105

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

7 Cut Sets and Cut Vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1157.1 Cut Sets and Fundamental Cut Sets . . . . . . . . . . . . . . . . . . . . . 115

7.1.1 Cut Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1157.1.2 Fundamental Cut Set (or Basic Cut Set) . . . . . . . . . . . . 116

7.2 Cut Vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1167.2.1 Cut Set with respect to a Pair of Vertices . . . . . . . . . . . 117

7.3 Separable Graph and its Block . . . . . . . . . . . . . . . . . . . . . . . . 1187.3.1 Separable Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1187.3.2 Block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7.4 Edge Connectivity and Vertex Connectivity . . . . . . . . . . . . . . . 1197.4.1 Edge Connectivity of a Graph . . . . . . . . . . . . . . . . . . . 1197.4.2 Vertex Connectivity of a Graph . . . . . . . . . . . . . . . . . . 119

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

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8 Coloring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1258.1 Properly Colored Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1258.2 Chromatic Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1268.3 Chromatic Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

8.3.1 Chromatic Number Obtained byChromatic Polynomial . . . . . . . . . . . . . . . . . . . . . . . . . 127

8.3.2 Chromatic Polynomial of a Graph G . . . . . . . . . . . . . . . 1288.4 Edge Contraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

9 Planar and Dual Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1359.1 Plane and Planar Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

9.1.1 Plane Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1359.1.2 Planar Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

9.2 Nonplanar Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1369.3 Embedding and Region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

9.3.1 Embedding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1369.3.2 Plane Representation . . . . . . . . . . . . . . . . . . . . . . . . . . 137

9.4 Regions or Faces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1379.5 Kuratowski’s Two Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

9.5.1 Kuratowski’s First Graph . . . . . . . . . . . . . . . . . . . . . . . 1389.5.2 Kuratowski’s Second Graph . . . . . . . . . . . . . . . . . . . . . 138

9.6 Euler’s Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1389.7 Edge Contractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1439.8 Subdivision, Branch Vertex, and Topological Minors. . . . . . . . . 1439.9 Kuratowi’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1469.10 Dual of a Planar Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

9.10.1 To Find the Dual of the Given Graph . . . . . . . . . . . . . . 1499.10.2 Relationship Between a Graph and its Dual Graph . . . . . 151

9.11 Edge Coloring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1539.11.1 k-Edge Colorable . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1539.11.2 Edge-Chromatic Number . . . . . . . . . . . . . . . . . . . . . . . 153

9.12 Coloring Planar Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1549.12.1 The Four Color Theorem . . . . . . . . . . . . . . . . . . . . . . . 1549.12.2 The Five Color Theorem . . . . . . . . . . . . . . . . . . . . . . . 155

9.13 Map Coloring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

10 Network Flows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15910.1 Transport Networks and Cuts . . . . . . . . . . . . . . . . . . . . . . . . . 159

10.1.1 Transport Network . . . . . . . . . . . . . . . . . . . . . . . . . . . 15910.1.2 Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

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10.2 Max-Flow Min-Cut Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 16210.3 Residual Capacity and Residual Network . . . . . . . . . . . . . . . . . 165

10.3.1 Residual Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16510.3.2 Residual Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

10.4 Ford-Fulkerson Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 16610.5 Ford-Fulkerson Algorithm with Modification

by Edmonds-Karp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16710.5.1 Time Complexity of Ford-Fulkerson Algorithm . . . . . . . 16710.5.2 Edmonds-Karp Algorithm . . . . . . . . . . . . . . . . . . . . . . 167

10.6 Maximal Flow: Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 17510.6.1 Multiple Sources and Sinks . . . . . . . . . . . . . . . . . . . . . 17510.6.2 Maximum Bipartite Matching. . . . . . . . . . . . . . . . . . . . 175

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

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About the Author

Dr. S. Saha Ray is currently working as an Associate Professor at the Departmentof Mathematics, National Institute of Technology, Rourkela, India. Dr. Saha Raycompleted his Ph.D. in 2008 from Jadavpur University, India. He received hisMCA degree in the year 2001 from Bengal Engineering College, Sibpur, Howrah,India. He completed his M.Sc. in Applied Mathematics at Calcutta University in1998 and B.Sc. (Honors) in Mathematics at St. Xavier’s College, Kolkata, in 1996.Dr. Saha Ray has about 12 years of teaching experience at undergraduate andpostgraduate levels. He also has more than 10 years of research experience invarious field of Applied Mathematics. He has published several research papers innumerous fields and various international journals of repute like TransactionASME Journal of Applied Mechanics, Annals of Nuclear Energy, Physica Scripta,Applied Mathematics and Computation, and so on. He is a member of the Societyfor Industrial and Applied Mathematics (SIAM) and American MathematicalSociety (AMS). He was the Principal Investigator of the BRNS research projectgranted by BARC, Mumbai. Currently, he is acting as Principal Investigator of aresearch Project financed by DST, Govt. of India. It is not out of place to mentionthat he had been invited to act as lead guest editor in the journal entitledInternational Journal of Differential equations of Hindawi Publishing Corpora-tion, USA.

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