Tolerance graphs【2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载】

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1 Introduction1

1.1 Background and motivation1

1.2 Intersection graphs and interval graphs4

1.3 Tolerance graphs： definitions and examples5

1.4 Chordal， comparability， interval graphs7

1.5 Ordered sets13

1.6 The hierarchy of permutation， parallelogram， trapezoid，function and AT-free graphs15

1.7 Other families of graphs20

1.8 Other reading and general references24

1.9 Exercises25

2 Early work on tolerance graphs29

2.1 Notation and observations29

2.2 Permutation graphs and interval graphs31

2.3 Bounded tolerance graphs33

2.4 Tolerance graphs are weakly chordal36

2.5 Tolerance graphs are perfect40

2.6 A first look at unit vs.proper45

2.7 Classes of perfect graphs48

2.8 Exercises52

3 Trees， cotrees and bipartite graphs53

3.1 Trees and cotrees53

3.2 Bipartite tolerance graphs - the bounded case60

3.3 Exercises61

4 Interval probe graphs and sandwich problems63

4.1 Physical mapping of DNA63

4.2 Interval probe graphs65

4.3 The hierarchy of interval， probe， and tolerance graphs66

4.4 The trees that are interval probe graphs71

4.5 Partitioned interval probe graphs73

4.6 The enhancement of a partitioned probe graph is chordal74

4.7 The Interval Graph Sandwich Problem77

4.8 The NP-completeness of the Interval Probe Graph Sandwich Problem80

4.9 Exercises82

5 Bitolerance and the ordered sets perspective84

5.1 The concept of a bounded tolerance order84

5.2 Classes of bounded bitolerance orders85

5.3 Geometric interpretations91

5.4 Exercises96

6 Unit and 50% tolerance orders98

6.1 Unit tolerance orders with six or fewer elements98

6.2 Unit vs.proper for bounded bitolerance orders103

6.3 Width 2 bounded tolerance orders107

6.4 Exercises108

7 Comparability invariance results109

7.1 Comparability invariance109

7.2 Autonomous sets and Gallai’s Theorem111

7.3 Dimension is a comparability invariant112

7.4 Bounded tolerance orders113

7.5 Unit bitolerance and unit tolerance orders115

7.6 Exercises122

8 Recognition of bounded bitolerance orders and trapezoid graphs124

8.1 Preliminaries125

8.2 The order B（I） of extreme corners127

8.3 The isomorphism between B（P） and B（I*）130

8.4 The recognition algorithm and its complexity132

8.5 Exercises133

9 Algorithms on tolerance graphs135

9.1 Tolerance and bounded tolerance representations136

9.2 Coloring tolerance representations137

9.3 Maximum weight stable set of a tolerance representation140

9.4 Exercises144

10 The hierarchy of classes of bounded bitolerance orders146

10.1 Introduction146

10.2 Equivalent classes148

10.3 Bipartite orders152

10.4 Separating examples158

10.5 Exercises163

11 Tolerance models of paths and subtrees of a tree164

11.1 Introduction164

11.2 Intersection models164

11.3 Discrete models165

11.4 Neighborhood subtrees169

11.5 Neighborhood subtree tolerance （NeST） graphs173

11.6 Subclasses of NeST graphs176

11.7 The hierarchy of NeST graphs183

11.8 A connection with threshold and threshold tolerance graphs187

11.9 Exercises191

12 φ-tolerance graphs193

12.1 Introduction193

12.2 φ-tolerance chain graphs195

12.3 Archimedean φ-tolerance graphs201

12.4 Polynomial functions209

12.5 Every graph can be represented by an Archimedean polynomial210

12.6 Construction of a universal Archimedean tolerance function213

12.7 Unit and proper representations215

12.8 Exercises217

13 Directed tolerance graphs219

13.1 Ferrers dimension 2220

13.2 Bounded bitolerance digraphs222

13.3 Recognition of bounded bitolerance digraphs224

13.4 Characterizations of bounded bitolerance digraphs225

13.5 The digraph hierarchy228

13.6 Cycles234

13.7 Trees237

13.8 Unit vs.proper243

13.9 Exercises248

14 Open questions and further directions of research249

References253

Index of symbols260

Index262