Graph Theory and Applications

Transcription

1 Graph Theory and Applications with Exercises and Problems Jean-Claude Fournier WILEY

2 Table of Contents Introduction 17 Chapter 1. Basic Concepts The origin of the graph concept Definition of graphs Notation Representation Terminology Isomorphism and unlabeled graphs Planar graphs Complete graphs Subgraphs Customary notation Paths and cycles Paths Cycles Paths and cycles as graphs Degrees Regular graphs Connectedness 35 7

3 8 Graph Theory and Applications 1.7 Bipartite graphs Characterization Algorithmic aspects Representations of graphs inside a machine Weighted graphs Exercises 41 Chapter 2. Trees Definitions and properties First properties of trees Forests Bridges Tree characterizations Spanning trees An interesting illustration of trees Spanning trees in a weighted graph Application: minimum spanning tree problem The problem Kruskal's algorithm Justification Implementation Complexity Connectivity Block decomposition fc-connectivity fc-connected graphs Menger's theorem Edge connectivity 63

4 Table of Contents fc-edge-connected graphs Application to networks Hypercube Exercises 66 Chapter 3. Colorings Coloring problems Edge coloring Basic results Algorithmic aspects The timetabling problem Room constraints An example Conclusion Exercises 81 Chapter 4. Directed Graphs Definitions and basic concepts Notation Terminology Representation Underlying graph "Directed" concepts Indegrees and outdegrees Strongly connected components Representations of digraphs inside a machine Acyclic digraphs Acyclic numbering 90

5 10 Graph Theory and Applications Characterization Practical aspects Arborescences Drawings Terminology Characterization of arborescences Subarborescences Ordered arborescences Directed forests Exercises 95 Chapter 5. Search Algorithms Depth-first search of an arborescence Iterative form Visits to the vertices Justification Complexity Optimization of a sequence of decisions The eight queens problem Application to game theory: finding a winning strategy Associated arborescence Example The minimax algorithm Implementation In concrete terms Pruning Depth-first search of a digraph Comments 110

6 Table of Contents Justification Complexity Extended depth-first search Justification Complexity Application to acyclic numbering Acyclic numbering algorithms Practical implementation Exercises 117 Chapter 6. Optimal Paths Distances and shortest paths problems A few definitions Types of problems Case of non-weighted digraphs: breadth-first search Application to calculation of distances Justification and complexity Determining the shortest paths Digraphs without circuits Shortest paths Longest paths Formulas Application to scheduling Potential task graph Earliest starting times Latest starting times Total slacks and critical tasks Free slacks 131

7 12 Graph Theory and Applications More general constraints Practical implementation Positive lengths Justification Associated shortest paths Implementation and complexity Undirected graphs Other cases Floyd's algorithm Exercises 143 Chapter 7. Matchings Matchings and alternating paths A few definitions Concept of alternating paths and Berge's theorem Matchings in bipartite graphs Matchings and transversals Assignment problem The Hungarian method Justification Concept of alternating trees Complexity Maximum matching algorithm Justification Complexity Optimal assignment problem Kuhn-Munkres algorithm Justification 168

8 Table of Contents Complexity Exercises 171 Chapter 8. Flows Flows in transportation networks Interpretation Single-source single-sink networks The max-flow min-cut theorem Concept of unsaturated paths Maximum flow algorithm Justification Complexity Flow with stocks and demands Revisiting theorems Menger's theorem Hall's theorem König's theorem Exercises 194 Chapter 9. Euler Tours Euler trails and tours Principal result Algorithms Example Complexity Elimination of recursion The Rosenstiehl algorithm The Chinese postman problem 207

9 14 Graph Theory and Applications The Edmonds-Johnson algorithm Complexity Example Exercises 212 Chapter 10. Hamilton Cycles Hamilton cycles A few simple properties The traveling salesman problem Complexity of the problem Applications Approximation of a difficult problem Concept of approximate algorithms Approximation of the metric TSP An approximate algorithm Justification and evaluation Amelioration Christofides' algorithm Justification and evaluation Another approach Upper and lower bounds for the optimal value Exercises 234 Chapter 11. Planar Representations Planar graphs Euler's relation Characterization of planar graphs Algorithmic aspect 242

10 Table of Contents Other properties of planar graphs Other graph representations Minimum crossing number Thickness Exercises 244 Chapter 12. Problems with Comments Problem 1: A proof of ^-connectivity Problem Comments Problem 2: An application to compiler theory Problem Comments Problem 3: Kernel of a digraph Problem Comments Problem 4: Perfect matching in a regular bipartite graph Problem Comments Problem 5: Birkhoff-Von Neumann's theorem Problem Comments Problem 6: Matchings and tilings Problem Comments Problem 7: Strip mining Problem Comments 259

11 16 Graph Theory and Applications Appendix A. Expression of Algorithms 261 A.l Algorithm 262 A.2 Explanations and commentaries 262 A.3 Other algorithms 265 A.4 Comments 265 Appendix B. Bases of Complexity Theory 267 B.l The concept of complexity 267 B.2 Class P 269 B.3 Class NP 272 B.4 NP-complete problems 273 B.5 Classification of problems 274 B.6 Other approaches to difficult problems 276 Bibliography 277 Index 279