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Chapter Ⅰ Graphs and Subgraphs1

Ⅰ.1 Definitions1

Ⅰ.2 Isomorphism5

Ⅰ.3 Subgraphs9

Ⅰ.4 Vertices of attachment11

Ⅰ.5 Components and connection14

Ⅰ.6 Deletion of an edge18

Ⅰ.7 Lists of nonisomorphic connected graphs22

Ⅰ.8 Bridges27

Ⅰ.9 Notes30

Exercises31

References31

Chapter Ⅱ Contractions and the Theorem of Menger32

Ⅱ.1 Contractions32

Ⅱ.2 Contraction of an edge37

Ⅱ.3 Vertices of attachment41

Ⅱ.4 Separation numbers43

Ⅱ.5 Menger's Theorem46

Ⅱ.6 Hall's Theorem50

Ⅱ.7 Notes52

Exercises53

References53

Chapter Ⅲ 2-Connection54

Ⅲ.1 Separable and 2-connected graphs54

Ⅲ.2 Constructions for 2-connected graphs56

Ⅲ.3 Blocks60

Ⅲ.4 Arms64

Ⅲ.5 Deletion and contraction of an edge66

Ⅲ.6 Notes68

Exercises68

References69

Chapter Ⅳ 3-Connection70

Ⅳ.1 Multiple connection70

Ⅳ.2 Some constructions for 3-connected graphs74

Ⅳ.3 3-blocks83

Ⅳ.4 Cleavages95

Ⅳ.5 Deletions and contractions of edges104

Ⅳ.6 The Wheel Theorem111

Ⅳ.7 Notes113

Exercises114

References114

Chapter Ⅴ Reconstruction115

Ⅴ.1 The Reconstruction Problem115

Ⅴ.2 Theory and practice118

Ⅴ.3 Kelly's Lemma119

Ⅴ.4 Edge-reconstruction122

Ⅴ.5 Notes123

Exercises124

References124

Chapter Ⅵ Digraphs and Paths125

Ⅵ.1 Digraphs125

Ⅵ.2 Paths129

Ⅵ.3 The BEST Theorem133

Ⅵ.4 The Matrix-Tree Theorem138

Ⅵ.5 The Laws of Kirchhoff142

Ⅵ.6 Identification of vertices149

Ⅵ.7 Transportation Theory152

Ⅵ.8 Notes158

Exercises159

References159

Chapter Ⅶ Alternating Paths161

Ⅶ.1 Cursality161

Ⅶ.2 The bicursal subgraph163

Ⅶ.3 Bicursal units167

Ⅶ.4 Alternating barriers168

Ⅶ.5 f-factors and f-barriers170

Ⅶ.6 The f-factor theorem174

Ⅶ.7 Subgraphs of minimum deficiency178

Ⅶ.8 The bipartite case180

Ⅶ.9 A theorem of Erd？s and Gallai181

Ⅶ.10 Notes183

Exercises183

References184

Chapter Ⅷ Algebraic Duality185

Ⅷ.1 Chain-groups185

Ⅷ.2 Primitive chains188

Ⅷ.3 Regular chain-groups194

Ⅷ.4 Cycles197

Ⅷ.5 Coboundaries200

Ⅷ.6 Reductions and contractions204

Ⅷ.7 Algebraic duality206

Ⅷ.8 Connectivity209

Ⅷ.9 On transportation theory215

Ⅷ.10 Incidence matrices217

Ⅷ.11 Matroids218

Ⅷ.12 Notes219

Exercises219

References220

Chapter Ⅸ Polynomials Associated with Graphs221

Ⅸ.1 V-functions221

Ⅸ.2 The chromatic polynomial226

Ⅸ.3 Colorings of graphs233

Ⅸ.4 The flow polynomial237

Ⅸ.5 Tait colorings240

Ⅸ.6 The dichromate of a graph243

Ⅸ.7 Some remarks on reconstruction248

Ⅸ.8 Notes250

Exercises251

References251

Chapter Ⅹ Combinatorial Maps253

Ⅹ.1 Definitions and preliminary theorems253

Ⅹ.2 Orientability257

Ⅹ.3 Duality259

Ⅹ.4 Isomorphism261

Ⅹ.5 Drawings of maps263

Ⅹ.6 Angles268

Ⅹ.7 Operations on maps268

Ⅹ.8 Combinatorial surfaces275

Ⅹ.9 Cycles and coboundaries281

Ⅹ.10 Notes283

Exercises284

References284

Chapter Ⅺ Planarity285

Ⅺ.1 Planar graphs285

Ⅺ.2 Spanning subgraphs288

Ⅺ.3 Jordan's Theorem290

Ⅺ.4 Connectivity in planar maps296

Ⅺ.5 The cross-cut Theorem306

Ⅺ.6 Bridges311

Ⅺ.7 An algorithm for planarity314

Ⅺ.8 Peripheral circuitsin 3-connected graphs317

Ⅺ.9 Kuratowski's Theorem320

Ⅺ.10 Notes325

Exercises325

References326

Index327