Handbook of Product Graphs

ISBN: 9781138199088 出版年：2011 页码：518 Hammack, Richard Imrich, Wilfried Klavzar, Sandi CRC Press

A BRIEF INTRODUCTION TO GRAPHS AND THEIR PRODUCTS Graphs Graphs and Subgraphs Paths and Cycles Trees and Forests Planar Graphs Automorphisms and Invariants Automorphisms Vertex-Transitivity Graph Invariants The No-Homomorphism Lemma Hypercubes and Isometric Subgraphs Hypercubes are Sparse Isometric Subgraphs Median Graphs Retracts Graph Products Three Fundamental Products Commutativity, Associativity, and Multiple Factors Projections and Layers Classification of Products The Four Standard Graph Products The Cartesian Product The Strong Product The Direct Product The Lexicographic Product FACTORIZATION AND CANCELLATION Cartesian Product Prime Factor Decompositions Cartesian Product and Its Group Transitive Group Action on Products Cancellation S-Prime Graphs Strong Product Basic Properties and S-Thin Graphs Cliques and the Extraction of Complete Factors Unique Prime Factorization for Connected Graphs Automorphisms Direct Product Nonuniqueness of Prime Factorization R-Thin Graphs The Cartesian Skeleton Factoring Connected, Nonbipartite, R-Thin Graphs Factoring Connected, Nonbipartite Graphs Automorphisms Applications to the Strong Product Cancellation Cancellation for the Strong Product Cancellation for the Direct Product Anti-Automorphisms and Factorials Graph Exponentiation Lexicographic Product Basic Properties Self-Complementarity and Cancellation Properties Commutativity Factorizations and Nonuniqueness Automorphisms ISOMETRIC EMBEDDINGS The Relation THETA and Partial Cubes Definition and Basic Properties of THETA Characterizations of Partial Cubes Cubic Partial Cubes Scale Embeddings into Hypercubes Median Graphs Mulder's Convex Expansion Inequalities for Median Graphs and Partial Cubes Median Graphs as Retracts A Fixed Cube Theorem Median Networks in Human Genetics The Canonical Isometric Embedding The Embedding and Its Properties The Relation THETA and the Cartesian Product Automorphisms of Canonical Embeddings A Dynamic Location Problem Hamming Graphs Graphs with Finite Windex Quasi-Median Graphs and Generalizations Graphs with Finite Windex are Quasi-Median Graphs Isometries in Strong Products and Product Dimensions Strong Isometric Dimension Retracts of Strong Products Other Product Graph Dimensions Fixed Box Theorems Gated Subgraphs and Median Functions A Fixed Box Theorem for Median Function-Closed Graphs Feder-Tardif's Fixed Box Theorems Fixed Points of Several Nonexpansive Mappings ALGORITHMS Graph Representation and Algorithms Time and Space Complexity Adjacency List Breadth-First Search Adjacency Matrix Recognizing Hypercubes and Partial Cubes Hypercubes Partial Cubes Efficient Computation of THETA* Recognizing Partial Cubes in Quadratic Time Chemical Graphs and the Wiener Index Benzenoid Graphs as Partial Cubes The Wiener Index of Benzenoid Graphs in Linear Time The Wiener Index via the Canonical Isometric Embedding Arboricity, Squares, and Triangles Arboricity Listing Squares and Triangles Recognizing Median Graphs A Simple Algorithm A Fast Algorithm Triangle-Free Graphs and Median Graphs Recognizing Partial Hamming Graphs and Quasi-Median Graphs Hamming Graphs and Partial Hamming Graphs Quasi-Median Graphs Computing the Windex Factoring the Cartesian Product Product Relation A Simple Algorithm Coordinatization Factorization in O(m log n) Time Factorization in Linear Time and Space Recognizing Direct, Strong, and Lexicographic Products Direct Product Strong Product Factoring Thin Graphs Factoring Non-Thin Graphs Lexicographic Product INVARIANTS Connectivity Cartesian Product Critically Connected Graphs and the Lexicographic Product Strong and Direct Products Coloring and Hedetniemi's Conjecture Product Coloring Bounds and Three Applications Fractional and Circular Chromatic Number Hedetniemi's Conjecture Hedetniemi's Conjecture for 4-Chromatic Graphs Circular and Fractional Version of Hedetniemi's Conjecture Independence Number and Shannon Capacity Shannon Capacity Independence in Direct Products Independence in Cartesian Products Domination and Vizing's Conjecture Vizing's Conjecture Clark and Suen's Approach Fractional Version of Vizing's Conjecture Domination in Direct Products Cycle Spaces and Bases The Cycle Space of a Graph Minimum Cycle Bases for Cartesian and Strong Products Minimum Cycle Bases for the Lexicographic Product Minimum Cycle Bases for the Direct Product Selected Results One-Factorization and Edge-Coloring Hamilton Cycles and Hamiltonian Decompositions Clique Minors in Cartesian Products Reconstruction, Topological Embeddings, and Flows Modeling Complex Networks RELATED CONCEPTS Infinite Graphs Growth Rate and Ends Free Product Transitive Median Graphs with Finite Blocks Two-Ended Median Graphs Cartesian Product Strong and Direct Product Lexicographic Product Products of Digraphs Definitions Connectedness Tournaments and the Lexicographic Product Prime Factorings Cancellation Near Products Graph Bundles Approximate Graph Products Graph Spectra Zig-Zag Product Appendix: Hints and Solutions to Exercises Bibliography Author Index Subject Index Symbol Index