This is a huge book, almost 200 pages longer than the already massive first edition. One is tempted to call it "Everything You Wanted to Know about Graph Theory but Were Afraid to Ask". Nonetheless, Graph Theory and Its Applications is a very good textbook.
What makes it good is strong rapport with the reader, a coherent organization, and consistently clear exposition. The book is aimed at a diverse set of readers. Courses based on this book could be directed toward computer science (concentrating on data structures and algorithms), operations research (focusing on discrete optimization), or mathematics (emphasizing the algebraic and topological aspects). The text is most appropriate for advanced undergraduates or beginning graduate students. Since it is essentially self-contained, it could also be used profitably for self-study.
The basics of graph theory (structure and representation, trees, connectivity, optimal transversal of graphs) are covered well in the first six chapters. Planarity and Kuratowski's theorem are discussed in depth in Chapter 7 where the authors begin their treatment of topological graph theory. There is a strong chapter on network flows and applications, and another on digraph models. Later chapters take up more advanced topics (enumeration, voltage graphs, design of nonplanar layouts). New to this second edition of the text are treatments of Ramsey theory as well as random graphs, an especially hot topic now.
Notable attractive features of the text are breakout boxes with pseudo-code for all significant algorithms (as well as suggestions for specific software implementations), hundreds of examples of graphs carefully integrated with the text, a glossary of terms with each chapter (especially useful in this terminology-heavy field), and a ton of exercises — many with solutions or hints. My only real quibble is that the pages all seem too full. No doubt this was in effort to keep the number of pages down in an already large book. But one wishes for more white space.
Of course, there are practical applications too. If you are ever caught in a maze consisting of tunnels and rooms, use Tarry's algorithm (page 189) to escape, using a variation on depth-first search. Never backtrack through a tunnel unless there is no alternative, and never go through a tunnel a second time in the same direction.
Bill Satzer (email@example.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.
INTRODUCTION TO GRAPH MODELS
Graphs and Digraphs
Common Families of Graphs
Graph Modeling Applications
Walks and Distance
Paths, Cycles, and Trees
Vertex and Edge Attributes: More Applications
STRUCTURE AND REPRESENTATION
Graph Isomorphism Revised!
Automorphisms and Symmetry Moved and revised!
Some Graph Operations
Tests for Non-Isomorphism
More Graph Operations
TREES Reorganized and revised!
Characterizations and Properties of Trees
Rooted Trees, Ordered Trees, and Binary Trees
Huffman Trees and Optimal Prefix Codes
Counting Labeled Trees: Prüfer Encoding
Counting Binary Trees: Catalan Recursion
SPANNING TREES Reorganized and revised!
Depth-First and Breadth-First Search
Minimum Spanning Trees and Shortest Paths
Applications of Depth-First Search
Cycles, Edge Cuts, and Spanning Trees
Graphs and Vector Spaces
Matroids and the Greedy Algorithm
Vertex- and Edge-Connectivity
Constructing Reliable Networks
Max-Min Duality and Menger's Theorems
OPTIMAL GRAPH TRAVERSALS
Eulerian Trails and Tours
DeBruijn Sequences and Postman Problems
Hamiltonian Paths and Cycles
Gray Codes and Traveling Salesman Problems
PLANARITY AND KURATOWSKI'S THEOREM Reorganized and revised!
Planar Drawings and Some Basic Surfaces
Subdivision and Homeomorphism
Extending Planar Drawings
Algebraic Tests for Planarity
Crossing Numbers and Thickness
DRAWING GRAPHS AND MAPS Reorganized and revised!
The Topology of Low Dimensions
Mathematical Model for Drawing Graphs
Regular Maps on a Sphere
Imbeddings on Higher-Order Surfaces
Geometric Drawings of Graphs New!
MEASUREMENT AND MAPPINGS New Chapter!
Distance in Graphs New!
Domination in Graphs New!
Intersection Graphs New!
Linear Graph Mappings Moved and revised!
Modeling Network Emulation Moved and revised!
ANALYTIC GRAPH THEORY New Chapter!
Ramsey Graph Theory New!
Extremal Graph Theory New!
Random Graphs New!
SPECIAL DIGRAPH MODELS Reorganized and revised!
Directed Paths and Mutual Reachability
Digraphs as Models for Relations
Project Scheduling and Critical Paths
Finding the Strong Components of a Digraph
NETWORK FLOWS AND APPLICATIONS
Flows and Cuts in Networks
Solving the Maximum-Flow Problem
Flows and Connectivity
Matchings, Transversals, and Vertex Covers
GRAPHICAL ENUMERATION Reorganized and revised!
Automorphisms of Simple Graphs
Graph Colorings and Symmetry
Cycle-Index Polynomial of a Permutation Group
More Counting, Including Simple Graphs
ALGEBRAIC SPECIFICATION OF GRAPHS
Cayley Graphs and Regular Voltages
Symmetric Graphs and Parallel Architectures
NON-PLANAR LAYOUTS Reorganized and revised!
Representing Imbeddings by Rotations
Genus Distribution of a Graph
Voltage-Graph Specification of Graph Layouts
Non KVL Imbedded Voltage Graphs
Heawood Map-Coloring Problem
Relations and Functions
Some Basic Combinatorics
SOLUTIONS AND HINTS New!
Index of Applications
Index of Algorithms
Index of Notations