Graphs, Algorithms, and Optimization

Abstract

GRAPHS AND THEIR COMPLEMENTS Degree sequences Analysis PATHS AND WALKS Complexity Walks The shortest path problem Weighted graphs and Dijkstra's algorithm Data structures. Floyd's algorithm SOME SPECIAL CLASSES OF GRAPHS Bipartite graphs Line graphs Moore graphs Euler tours TREES AND CYCLES Fundamental Co-trees and bonds Spanning tree algorithms THE STRUCTURE OF TREES Non-rooted Read's tree encoding algorithm Generating rooted trees Generating non-rooted trees Prufer sequences Spanning trees The matrix-tree theorem CONNECTIVITY Blocks Finding the blocks of a graph The depth-first search ALTERNATING PATHS AND MATCHINGS. The Hungarian algorithm Perfect matchings and 1-factorizations The subgraph problem Coverings in bipartite graphs Tutte's theorem NETWORK FLOWS Introduction The Ford-Fulkerson algorithm Matchings and flows Menger's theorems Disjoint paths and separating sets Notes HAMILTON CYCLES The crossover algorithm The Hamilton closure The extended multi-path algorithm The traveling salesman problem The ?TSP Christofides' algorithm DIGRAPHS Activity graphs, Critical paths Topological order Strong components An application to fabrics Tournaments Satisfiability GRAPH COLORINGS Cliques Mycielski's construction Critical graphs Chromatic polynomials Edge colorings NP-completeness PLANAR GRAPHS Jordan curves Graph minors Subdivisions Euler's formula Rotation systems Dual graphs Platonic solids Triangulations The sphere 5 Whitney's theorem Medial digraphs The 4-color problem Straight line drawings Kuratowski's theorem The Hopcroft-Tarjan Algorithm GRAPHS AND SURFACES Surfaces Graph embeddings Graphs on the torus Graphs on the projective plane LINEAR PROGRAMMING The simplex algorithm Cycling THE PRIMAL-DUAL ALGORITHM Alternate form of the primal and its dual Geometric interpretation Complementary slackness The dual of the shortest path problem The primal-dual algorithm DISCRETE LINEAR PROGRAMMING Backtracking Branch and bound Unimodular matrices BIBLIOGRAPHY INDEX