Graph Theory and Its Applications

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.

Structure and Representation

Graph Isomorphism. Automorphism and Symmetry. Subgraphs. Some Graph Operations. Tests for Non-Isomorphism. Matrix Representation. More Graph Operations.

Trees

Characterizations and Properties of Trees. Rooted Trees, Ordered Trees, and Binary Trees. Binary-Tree Traversals. Binary-Search Trees. Huffman Trees and Optimal Prefix Codes. Priority Trees. Counting Labeled Trees. Counting Binary Trees.

Spanning Trees

Tree Growing. 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.

Connectivity

Vertex and Edge-Connectivity. Constructing Reliable Networks. Max-Min Duality and Menger's Theorems. Block Decompositions.

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

Planar Drawings and Some Basic Surfaces. Subdivision and Homeomorphism. Extending Planar Drawings. Kuratowski's Theorem. Algebraic Tests for Planairty. Planarity Algorithm. Crossing Numbers and Thickness.

Graph Colorings

Vertex-Colorings. Map-Colorings. Edge-Colorings. Factorization.

Special Digraph Models

Directed Paths and Mutual Reachability. Digraphs as Models for Relations. Tournaments. Project Scheduling. 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.

Graph Colorings and Symmetry

Automorphisms of Simple Graphs. Equivalence Classes of Colorings.

Appendix