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Overview

This book introduces polyhedra as a tool for graph theory and discusses their properties and applications in solving the Gauss crossing problem. The discussion is extended to embeddings on manifolds, particularly to surfaces of genus zero and non-zero via the joint tree model, along with solution algorithms. Given its rigorous approach, this book would be of interest to researchers in graph theory and discrete mathematics.

Product Details

ISBN-13: 9783110476699
Publisher: De Gruyter
Publication date: 03/06/2017
Pages: 369
Product dimensions: 6.69(w) x 9.45(h) x (d)
Age Range: 18 Years

About the Author

Yanpei Liu, Beijing Jiaotong University, Beijing, China.

Table of Contents

1 Preliminaries 1

1.1 Sets and relations 1

1.2 Partitions and permutations 5

1.3 Graphs and networks 9

1.4 Groups and spaces 15

1.5 Notes 20

2 Polyhedra 22

2.1 Polygon double covers 22

2.2 Supports and skeletons 25

2.3 Orientable polyhedra 28

2.4 Non-orientable polyhedra 30

2.5 Classic polyhedra 32

2.6 Notes 34

3 Surfaces 35

3.1 Polyhegons 35

3.2 Surface closed curve axiom 39

3.3 Topological transformations 42

3.4 Complete invariants 46

3.5 Graphs on surfaces 48

3.6 Up-embeddability 52

3.7 Notes 56

4 Homology on Polyhedra 58

4.1 Double cover by travels 58

4.2 Homology 60

4.3 Cohomology 65

4.4 Bicycles 70

4.5 Notes 75

5 Polyhedra on the Sphere 78

5.1 Planar polyhedra 78

5.2 Jordan closed-curve axiom 84

5.3 Uniqueness 87

5.4 Straight-line representations 91

5.5 Convex representation 93

5.6 Notes 95

6 Automorphisms of a Polyhedron 97

6.1 Automorphisms of polyhedra 97

6.2 Eulerian and non-Eulerian codes 102

6.3 Determination of automorphisms 110

6.4 Asymmetrization 124

6.5 Notes 127

7 Gauss Crossing Sequences 129

7.1 Crossing polyhegons 129

7.2 Dehn's transformation 133

7.3 Algebraic principles 137

7.4 Gauss crossing problem 141

7.5 Notes 143

8 Cohomology on Graphs 145

8.1 Immersions 145

8.2 Realization of planarity 148

8.3 Reductions 151

8.4 Planarity auxiliary graphs 154

8.5 Basic conclusions 158

8.6 Notes 163

9 Embeddability on Surfaces 165

9.1 Joint tree model 165

9.2 Associate polybegons 167

9.3 A transformation 168

9.4 Criteria of embeddability 171

9.5 Notes 173

10 Embeddings on Sphere 175

10.1 Left and right determinations 175

10.2 Forbidden configurations 179

10.3 Basic order characterization 185

10.4 Number of planar embeddings 192

10.5 Notes 197

11 Orthogonality on Surfaces 198

11.1 Definitions 198

11.2 On surfaces of genus zero 205

11.3 Surface models 225

11.4 On surfaces of genus not zero 228

11.5 Notes 229

12 Net Embeddings 231

12.1 Definitions 231

12.2 Face admissibility 236

12.3 General criterion 242

12.4 Special criteria 248

12.5 Notes 255

13 Extremality on Surfaces 257

13.1 Maximal genus 257

13.2 Minimal genus 261

13.3 Shortest embedding 264

13.4 Thickness 273

13.5 Crossing number 275

13.6 Minimal bend 277

13.7 Minimal area 283

13.8 Notes 288

14 Matroidal Graphicness 291

14.1 Definitions 291

14.2 Binary matroids 292

14.3 Regularity 295

14.4 Graphicness 300

14.5 Cographicness 305

14.6 Notes 306

15 Knot Polynomials 308

15.1 Definitions 308

15.2 Knot diagram 312

15.3 Tutte polynomial 317

15.4 Pan-polynomial 320

15.5 Jones Polynomial 327

15.6 Notes 329

Bibliography 331

Subject Index 347

Author Index 355

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