Graphs Theory and Applications: With Exercises and Problems

Introduction 17

Chapter 1. Basic Concepts 21

1.1 The origin of the graph concept 21

1.2 Definition of graphs 24

1.3 Subgraphs 28

1.4 Paths and cycles 29

1.5 Degrees 33

1.6 Connectedness 35

1.7 Bipartite graphs 36

1.8 Algorithmic aspects 37

1.9 Exercises 41

Chapter 2. Trees 45

2.1 Definitions and properties 45

2.2 Spanning trees 49

2.3 Application: minimum spanning tree problem 54

2.4 Connectivity 59

2.5 Exercises 66

Chapter 3. Colorings 71

3.1 Coloring problems 71

3.2 Edge coloring 71

3.3 Algorithmic aspects 73

3.4 The timetabling problem 75

3.5 Exercises 81

Chapter 4. Directed Graphs 83

4.1 Definitions and basic concepts 83

4.2 Acyclic digraphs 90

4.3 Arborescences 92

4.4 Exercises 95

Chapter 5. Search Algorithms 97

5.1 Depth-first search of an arborescence 97

5.2 Optimization of a sequence of decisions 103

5.3 Depth-first search of a digraph 109

5.4 Exercises 117

Chapter 6. Optimal Paths 119

6.1 Distances and shortest paths problems 119

6.2 Case of non-weighted digraphs: breadth-first search 120

6.3 Digraphs without circuits 125

6.4 Application to scheduling 128

6.5 Positive lengths 134

6.6 Other cases 142

6.7 Exercises 143

Chapter 7. Matchings 149

7.1 Matchings and alternating paths 149

7.2 Matchings in bipartite graphs 152

7.3 Assignment problem 156

7.4 Optimal assignment problem 164

7.5 Exercises 171

Chapter 8. Flows 173

8.1 Flows in transportation networks 173

8.2 The max-flow min-cut theorem 177

8.3 Maximum flow algorithm 180

8.4 Flow with stocks and demands 188

8.5 Revisiting theorems 191

8.6 Exercises 194

Chapter 9. Euler Tours 197

9.1 Euler trails and tours 197

9.2 Algorithms 201

9.3 The Chinese postman problem 207

9.4 Exercises 212

Chapter 10. Hamilton Cycles 215

10.1 Hamilton cycles 215

10.2 The traveling salesman problem 218

10.3 Approximation of a difficult problem 220

10.4 Approximation of themetric TSP 223

10.5 Exercises 234

Chapter 11. Planar Representations 237

11.1 Planar graphs 237

11.2 Other graph representations 242

11.3 Exercises 244

Chapter 12. Problems with Comments 247

12.1 Problem 1: A proof of k-connectivity 247

12.2 Problem2: An application to compiler theory 249

12.3 Problem3: Kernel of a digraph 251

12.4 Problem 4: Perfect matching in a regular bipartite graph 253

12.5 Problem5: Birkhoff-Von Neumann’s theorem 254

12.6 Problem 6: Matchings and tilings 256

12.7 Problem7: Strip mining 258

Appendix A. Expression of Algorithms 261

Appendix B. Bases of Complexity Theory 267

Bibliography 277

Index 279