On the stable set polytope of claw-free graphs

Abstract

In this thesis we focus our attention on the stable set polytope of claw-free graphs. This problem has been open for many years and albeit all the efforts engaged during those last three years, it is still open. This does not mean that no progress has been made and we hope to the contrary that this thesis contains some important advances and that the reader will share this point of view. Understanding the stable set polytope requires seizing its geometry. But its most interesting facets appear in high dimension and thus one must develop a more algebraic than geometric intuition of the problem. In this thesis we try to convey the intuition we have developed. Indeed we provide several constructions and examples of facets for the stable set polytope of circulant graphs, quasi-line graphs and claw-free graphs. This intuition allowed us to prove some important results in the characterization of the stable set polytope. The main contribution of the thesis is without any doubt the proof of Ben Rebea conjecture leading to a characterization of the stable set polytope of quasi-line graphs. This result claims that clique, non-negativity and clique family inequalities are enough to characterize this polytope. It builds upon a recent decomposition theorem by Chudnovsky and Seymour for claw-free graphs. This is not the end of the story and we try to determine which of the inequalities are essential i.e. which are the facets of the polytope. We answer this question in various ways. First we propose a combinatorial characterization of the rank facets of the stable set polytope of fuzzy circular interval graphs. Then we prove that all the facets of the stable set polytope of quasi-line graphs can be obtained by lifting rank facets in lower dimension and thus we determine a strongly minimal characterization of the facets of this polytope. Finally we give strong necessary conditions for clique family inequalities to be facet producing in quasi-line graphs. It allows in particular to give a better characterization of the stable set polytope of circular interval graphs which proves a conjecture by Pecher and Wagler for the stable set polytope of circulants. Even if we mainly focus on polyhedral results, we also derive various algorithmic ones. We develop an algorithm to solve the weighted stable set problem in fuzzy circular interval graphs and another one to determine a maximum cardinality stable set in a quasi-line graph. We finally provide procedures to separate over certain valid inequalities for the stable set polytope of general graphs. We conclude the thesis with some perspectives. In particular, we formulate several conjectures about the stable set polytope of claw-free graphs and some possible lines of attack. We hope this will help researchers interested in continuing those works.