Graphs on Surfaces

Abstract

Contents: Chapter 1. Introduction Basic Definition Trees and bipartite graphs Blocks ConnectivityChapter 2. Planar Graphs Planar graphs and the Jordan Curve Theorem The Jordan-Schonflies Theorem The Theorem of Kuratowski Characterizations of planar graphs 3-connected planar graphs Dual graphs Planarity algorithms Circle packing representations The Riemann Mapping Theorem The Jordan Curve Theorem and Kuratowski's Theorem in general topological spacesChapter 3. Surfaces Classification of surfacesRotation systemsEmbedding schemesThe genus of a graphClassification of noncompact surfacesChapter 4. Embeddings Combinatorially, Contractibility, of Cycles, and the Genus Problem Embeddings combinatoriallyCycles of embedded graphsThe 3-path-conditionThe genus of a graphThe maximum genus of a graphChapter 5. The Width of Embeddings Edge-width 2-flippings and uniqueness of LEW-embeddings Triangulations Minimal triangulations of a given edge-width Face-width Minimal embeddings of a given face-width Embeddings of planar graphs The genus of a graph with a given nonorientable embedding Face-width and surface minors Face-width and embedding flexibility Combinatorial properties of embedded graphs of large widthChapter 6. Embedding Extensions and Obstructions Forbidden subgraphs and forbidden minors Bridges Obstruction in a bridge 2-restricted embedding extensions The forbidden subgraphs for the projective plane The minimal forbidden subgraphs for general surfacesChapter 7. Tree-Width and the Excluded Minor Theorem Tree-width and the excluded grid theoremThe excluded minor theorem for any fixed surfaceChapter 8. Colorings of Graphs on Surfaces Planar graphs are 5-choosable The Four Color Theorem Color critical graphs and the Heawood formula Coloring in a few colors Graphs without short cycles Appendix A. The minmal forbidden subgraphs for the projective plane Appendix B. The unavoidable configurations in planar triangulations Bibliography Index