Topological Theory of Graphs

Table of Content: Preface Chapter 1 Preliminaries 1.1 Sets and relations 1.2 Partitions and permutations 1.3 Graphs and networks 1.4 Groups and spaces 1.5 Notes Chapter 2 Polyhedra 2.1 Polygon double covers 2.2 Supports and skeletons 2.3 Orientable polyhedra 2.4 Nonorientable polyhedral 2.5 Classic polyhedral 2.6 Notes Chapter 3 Surfaces 3.1 Polyhegons 3.2 Surface closed curve axiom 3.3 Topological transformations 3.4 Complete invariants 3.5 Graphs on surfaces 3.6 Up-embeddability 3.7 Notes Chapter 4 Homology on Polyhedra 4.1 Double cover by travels 4.2 Homology 4.3 Cohomology 4.4 Bicycles 4.5 Notes Chapter 5 Polyhedra on the Sphere 5.1 Planar polyhedra 5.2 Jordan closed curve axiom 5.3 Uniqueness 5.4 Straight line representations 5.5 Convex representation 5.6 Notes Chapter 6 Automorphisms of a Polyhedron 6.1 Automorphisms 6.2 V -codes and F-codes 6.3 Determination of automorphisms 6.4 Asymmetrization 6.5 Notes Chapter 7 Gauss Crossing Sequences 7.1 Crossing polyhegons 7.2 Dehns transformation 7.3 Algebraic principles 7.4 Gauss Crossing problem 7.5 Notes Chapter 8 Cohomology on Graphs 8.1 Immersions 8.2 Realization of planarity 8.3 Reductions 8.4 Planarity auxiliary graphs 8.5 Basic conclusions 8.6 Notes Chapter 9 Embeddability on Surfaces 9.1 Joint tree model 9.2 Associate polyhegons 9.3 A transformation 9.4 Criteria of embeddability 9.5 Notes Chapter 10 Embeddings on the Sphere 10.1 Left and right determinations 10.2 Forbidden Congurations 10.3 Basic order characterization 10.4 Number of planar embeddings 10.5 Notes Chapter 11 Orthogonality on Surfaces 11.1 Denitions 11.2 On surfaces of genus zero 11.3 Surface Model 11.4 On surfaces of genus not zero 11.5 Notes Chapter 12 Net Embeddings 12.1 Denitions 12.2 Face admissibility 12.3 General criterion 12.4 Special criteria 12.5 Notes Chapter 13 Extremality on Surfaces 13.1 Maximal genus 13.2 Minimal genus 13.3 Shortest embedding 13.4 Thickness 13.5 Crossing number 13.6 Minimal bend 13.7 Minimal area 13.8 Notes Chapter 14 Matroidal Graphicness 14.1 Denitions 14.2 Binary matroids 14.3 Regularity 14.4 Graphicness 14.5 Cographicness 14.6 Notes Chapter 15 Knot Polynomials 15.1 Denitions 15.2 Knot diagram 15.3 Tutte polynomial 15.4 Pan-polynomial 15.5 Jones polynomial 15.6 Notes Reference Index