GRAPHS: THEORY AND ALGORITHMS
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1 GRAPHS: THEORY AND ALGORITHMS K. THULASIRAMAN M. N. S. SWAMY Concordia University Montreal, Canada A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York / Chichester / Brisbane / Toronto / Singapore
2 CONTENTS PREFACE 1 BASIC CONCEPTS 1.1 Some Basic Definitions / Subgraphs and Complements / Walks, Trails, Paths, and Circuits / Connectedness and Components of a Graph / Operations on Graphs / Special Graphs / Cut-Vertices and Separable Graphs / Isomorphism and 2-Isomorphism / Further Reading / Exercises / References / 29 2 TREES, CUTSETS, AND CIRCUITS 2.1 Trees, Spanning Trees, and Cospanning Trees / fc-trees, Spanning A:-Trees, and Forests / Rank and Nullity / Fundamental Circuits / Cutsets / Cuts / 43
3 VÜi CONTENTS 2.7 Fundamental Cutsets / Spanning Trees, Circuits, and Cutsets / Further Reading / Exercises / References / 54 3 EULERIAN AND HAMILTONIAN GRAPHS Eulerian Graphs / Hamiltonian Graphs / Further Reading / Exercises / References / 70 4 GRAPHS AND VECTOR SPACES Groups and Fields / Vector Spaces / Vector Space of a Graph / Dimensions of Circuit and Cutset Subspaces / Relationship between Circuit and Cutset Subspaces / Orthogonality of Circuit and Cutset Subspaces / Further Reading / Exercises / References / 96 5 DIRECTED GRAPHS Basic Definitions and Concepts / Graphs and Relations / Directed Trees or Arborescences / Directed Eulerian Graphs / Directed Spanning Trees and Directed Euler Trails / Directed Hamiltonian Graphs / Acyclic Directed Graphs / Tournaments / Further Reading / Exercises / References / 124
4 CONTENTS ix 6 MATRICES OF A GRAPH Incidence Matrix / Cut Matrix / Circuit Matrix / Orthogonality Relation / Submatrices of Cut, Incidence, and Circuit Matrices / Unimodular Matrices / The Number of Spanning Trees / The Number of Spanning 2-Trees / The Number of Directed Spanning Trees in a Directed Graph / Adjacency Matrix / The Coates and Mason Graphs / Further Reading / Exercises / References / PLANARITY AND DUALITY Planar Graphs / Euler's Formula / Kuratowski's Theorem and Other Characterizations of Planarity / Dual Graphs / Planarity and Duality / Further Reading / Exercises / References / CONNECTIVITY AND MATCHING Connectivity or Vertex Connectivity / Edge Connectivity / Graphs with Prescribed Degrees / Menger's Theorem / Matchings / Matchings in Bipartite Graphs / Matchings in General Graphs / Further Reading / Exercises / References / 234
5 X CONTENTS 9 COVERING AND COLORING Independent Sets and Vertex Covers / Edge Covers / Edge Coloring and Chromatic Index / Vertex Coloring and Chromatic Numbcr / Chromatic Polynomials / The Four-Color Problem / Further Reading / Exercises / References / MATROIDS Basic Definitions / Fundamental Properties / Equivalent Axiom Systems / Matroid Duality and Graphoids / Restriction, Contraction, and Minors of a Matroid / Representability of a Matroid / Binary Matroids / Orientable Matroids / Matroids and the Greedy Algorithm / Further Reading / Exercises / References / GRAPH ALGORITHMS Transitive Closure / Shortest Paths / Minimum Weight Spanning Tree / Optimum Branchings / Perfect Matching, Optimal Assignment, and Timetable Scheduling / The Chinese Postman Problem / Depth-First Search / Biconnectivity and Strong Connectivity / Reducibility of a Program Graph / ^-Numbering of a Graph / Planarity Testing / 373
6 CONTENTS Xi Further Reading / Exercises / References / FLOWS IN NETWORKS The Maximum Flow Problem / Maximum Flow Minimum Cut Theorem / Ford-Fulkerson Labeling Algorithm / Edmonds and Karp Modification of the Labeling Algorithm / Dinic Maximum Flow Algorithm / Maximal Flow in a Layered Network: The MPM Algorithm / Preflow Push Algorithm: Goldberg and Tarjan / Maximum Flow in 0-1 Networks / Maximum Matching in Bipartite Graphs / Menger's Theorems and Connectivities / NP-Completeness / Further Reading / Exercises / References / 439 AUTHOR INDEX 445 SUBJECT INDEX 451
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