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This book introduces polyhedra as a tool for graph theory and discusses their properties and applications in solving the Gauss crossing problem. Given its rigorous approach, this book would be of interest to researchers in graph theory and discrete mathematics.

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9783110476699

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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

Table of Content: PrefaceChapter 1 Preliminaries1.1 Sets and relations1.2 Partitions and permutations1.3 Graphs and networks1.4 Groups and spaces1.5 NotesChapter 2 Polyhedra2.1 Polygon double covers2.2 Supports and skeletons2.3 Orientable polyhedra2.4 Nonorientable polyhedral2.5 Classic polyhedral2.6 NotesChapter 3 Surfaces3.1 Polyhegons3.2 Surface closed curve axiom3.3 Topological transformations3.4 Complete invariants3.5 Graphs on surfaces3.6 Up-embeddability3.7 NotesChapter 4 Homology on Polyhedra4.1 Double cover by travels4.2 Homology4.3 Cohomology4.4 Bicycles4.5 NotesChapter 5 Polyhedra on the Sphere5.1 Planar polyhedra5.2 Jordan closed curve axiom5.3 Uniqueness5.4 Straight line representations5.5 Convex representation5.6 NotesChapter 6 Automorphisms of a Polyhedron6.1 Automorphisms6.2 V -codes and F-codes6.3 Determination of automorphisms6.4 Asymmetrization6.5 NotesChapter 7 Gauss Crossing Sequences7.1 Crossing polyhegons7.2 Dehns transformation7.3 Algebraic principles7.4 Gauss Crossing problem7.5 NotesChapter 8 Cohomology on Graphs8.1 Immersions8.2 Realization of planarity8.3 Reductions8.4 Planarity auxiliary graphs8.5 Basic conclusions8.6 NotesChapter 9 Embeddability on Surfaces9.1 Joint tree model9.2 Associate polyhegons9.3 A transformation9.4 Criteria of embeddability9.5 NotesChapter 10 Embeddings on the Sphere10.1 Left and right determinations10.2 Forbidden Congurations10.3 Basic order characterization10.4 Number of planar embeddings10.5 NotesChapter 11 Orthogonality on Surfaces11.1 Denitions11.2 On surfaces of genus zero11.3 Surface Model11.4 On surfaces of genus not zero11.5 NotesChapter 12 Net Embeddings12.1 Denitions12.2 Face admissibility12.3 General criterion12.4 Special criteria12.5 NotesChapter 13 Extremality on Surfaces13.1 Maximal genus13.2 Minimal genus13.3 Shortest embedding13.4 Thickness13.5 Crossing number13.6 Minimal bend13.7 Minimal area13.8 NotesChapter 14 Matroidal Graphicness14.1 Denitions14.2 Binary matroids14.3 Regularity14.4 Graphicness14.5 Cographicness14.6 NotesChapter 15 Knot Polynomials15.1 Denitions15.2 Knot diagram15.3 Tutte polynomial15.4 Pan-polynomial15.5 Jones polynomial15.6 NotesReferenceIndex

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

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Yanpei Liu, Beijing Jiaotong University, Beijing, China.

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