Product Graphs, Second Edition

Handbook of Product Graphs, Second Edition examines the dichotomy between the structure of products and their subgraphs. It also features the design of efficient algorithms that recognize products and their subgraphs and explores the relationship between graph parameters of the product and factors. Extensively revised and expanded, the handbook presents full proofs of many important results as well as up-to-date research and conjectures. Results and Algorithms New to the Second Edition: Cancellation results A quadratic recognition algorithm for partial cubes Results on the strong isometric dimension Computing the Wiener index via canonical isometric embedding Connectivity results A fractional version of Hedetniemi’s conjecture Results on the independence number of Cartesian powers of vertex-transitive graphs Verification of Vizing’s conjecture for chordal graphs Results on minimum cycle bases Numerous selected recent results, such as complete minors and nowhere-zero flows The second edition of this classic handbook provides a thorough introduction to the subject and an extensive survey of the field. The first three parts of the book cover graph products in detail. The authors discuss algebraic properties, such as factorization and cancellation, and explore interesting and important classes of subgraphs. The fourth part presents algorithms for the recognition of products and related classes of graphs. The final two parts focus on graph invariants and infinite, directed, and product-like graphs. Sample implementations of selected algorithms and other information are available on the book’s website, which can be reached via the authors’ home pages.

A BRIEF INTRODUCTION TO GRAPHS AND THEIR PRODUCTSGraphs 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 CANCELLATIONCartesian ProductPrime 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 EMBEDDINGSThe Relation T and Partial Cubes Definition and Basic Properties of TCharacterizations of Partial Cubes Cubic Partial Cubes Scale Embeddings into Hypercubes Median GraphsMulder’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 T 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 ALGORITHMSGraph Representation and AlgorithmsTime and Space Complexity Adjacency ListBreadth-First Search Adjacency Matrix Recognizing Hypercubes and Partial Cubes Hypercubes Partial Cubes Efficient Computation of T*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 INVARIANTSConnectivity Cartesian Product Critically Connected Graphs and the Lexicographic Product Strong and Direct Products Coloring and Hedetniemi’s Conjecture Product ColoringBounds 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 CONCEPTSInfinite GraphsGrowth 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

Richard Hammack is an associate professor in the Department of Mathematics and Applied Mathematics at Virginia Commonwealth University. Dr. Hammack is a member of the American Mathematical Society, the Mathematical Association of America, and the Institute of Combinatorics and its Applications. He earned a Ph.D. in mathematics from the University of North Carolina at Chapel Hill. Wilfried Imrich is professor emeritus in the Department of Mathematics and Information Technology at Montanuniversität Leoben. His research interests include the structure of finite and infinite graphs, graph automorphisms, combinatorial group theory, and graph algorithms. Dr. Imrich earned a Ph.D. from the University of Vienna. Sandi Klavžar is a professor in the Faculty of Mathematics and Physics at the University of Ljubljana and in the Faculty of Natural Sciences and Math at the University of Maribor. Dr. Klavžar is an editorial board member of Ars Mathematica Contemporanea, Asian-European Journal of Mathematics, Discussiones Mathematicae Graph Theory, European Journal of Combinatorics, and MATCH Communications in Mathematical and in Computer Chemistry.