本書不在于圖的拓撲性質(zhì)本身，而是著意以圖為代表的一些組合構(gòu)形為出發(fā)點，揭示與拓撲學(xué)中一些典型對蠏，如多面形、曲面、嵌入、紐結(jié)等的聯(lián)系，特別是顯示了定理有效化的途徑對于以拓撲學(xué)為代表的基礎(chǔ)數(shù)學(xué)的作用。同時，也提出了一些新的曲面模型，為超大規(guī)模集成電路的布線嘗試構(gòu)建多方面的理論基礎(chǔ)。
本書可作為基礎(chǔ)數(shù)學(xué)，應(yīng)用數(shù)學(xué)、系統(tǒng)科學(xué)、計算機科學(xué)等專業(yè)高年級本科生和研究生的補充教材，也可供相關(guān)專業(yè)的教師和科研工作者參考。
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 polyhedra
2.5 Classic polyhedra
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
5.5 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
Contents
Chapter 9 Embeddability on Surfaces
9.1 Joint tree model
9.2 Associate polyhegons
9.3 The exchanger
9.4 Criteria of embeddability
9.5 Notes
Chapter 10 Embeddings on the Sphere
10.1 Left and right determinations
10.2 Forbidden configurations
10.3 Basic order characterization
10.4 Number of planar embeddings
10.5 Notes
Chapter 11 Orthogonality on Surfaces
11.1 Definitions
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 Definitions
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 Definitions
14.2 Binary matroids
14.3 Regularity
14.4 Graphicness
14.5 Cographicness
14.6 Notes
Chapter 15 Knot Polynomials
15.1 Definitions
15.2 Knot diagram
15.3 Tutte polynomial
15.4 Pan-polynomial
15.5 Jones polynomial
15.6 Notes
Bibliography
Subject Index
Author Index
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