1 Partially ordered sets and graphs 15

Contents

Acknowledgements 9

Introduction 11

I Preliminaries 14

1 Partially ordered sets and graphs 15

1.1 Notations and conventions . . . . 15

1.2 Ordered sets . . . . 15

1.3 Particular classes of posets . . . . 18

1.4 Graphs . . . . 20

1.5 Comparability graphs . . . . 22

1.6 Linear extensions . . . . 24

1.7 Rooted trees and posets decomposition . . . . 31

2 Convexity 34 2.1 Convex sets and polytopes . . . . 34

2.2 Convex functions and convex programming . . . . 38

2.3 A word on linear programming . . . . 39

II Order polytopes 41 3 Order polytopes 42 3.1 Definitions and basic properties . . . . 42

3.2 Motivations . . . . 44

3.3 Axiomatic inequalities . . . . 45

3.4 Primary facets of order polytopes . . . . 46

3.5 Primary facets of P

. . . . 50

3.6 Primary facets of P

. . . . 51

3.7 Primary facets of P

and P

. . . . 53

1

4 Primary facets of the semiorder polytope 56

4.1 The lifting lemma . . . . 56

4.2 The primary valid inequalities for P

. . . . 58

4.3 The primary facet defining inequalities for P

. . . . 62

4.4 Links with the interval order polytope . . . . 71

4.5 Links with the weak order polytope . . . . 72

5 Linear extension polytopes 76 5.1 Definitions and basic properties . . . . 76

5.2 Comparability invariance . . . . 79

5.3 A linear relaxation for P

(P ) . . . . 81

5.4 Posets with extension dimension 2 . . . . 84

5.5 Connectivity of the incomparability graph . . . . 92

5.6 Reduction to particular posets of width 3 or 4 . . . . 99

5.7 Other extension polytopes . . . . 101

III Partial order entropy 103 6 Graph entropy 104 6.1 The stable set polytope . . . . 104

6.2 Definition and basic properties . . . . 105

6.3 Entropy of perfect graphs . . . . 107

6.4 Entropy of bipartite graphs . . . . 108

6.5 Graphs with high girth and high entropy . . . . 109

7 Poset entropy 111 7.1 Definition and motivations . . . . 111

7.2 A more intuitive definition of H(P ) . . . . 114

7.3 The posets achieving the bounds in inequality (7.1) . . . . 116

7.4 Posets of fixed width or fixed height . . . . 118

7.5 Asymptotic results . . . . 121

7.6 Substitutions . . . . 122

7.7 Posets with decomposition width at most 2 . . . . 124

8 Entropy of width-2 posets 127 8.1 Structure of the proof of Theorem 7.10 . . . . 127

8.2 The structure of G(P) and G(I (P )) . . . . 129

8.3 Phantom edges . . . . 132

8.4 Removing an incomparability with a small overlap . . . . 134

8.4.1 Removing a phantom edge . . . . 135

8.4.2 Removing an edge within an epoch . . . . 135

8.5 The final discussion . . . . 140

8.6 Special cases . . . . 141

2

8.7 Asymptotic behavior . . . . 144

Bibliography 146

Index 152

3