The stable set polytope of quasi-line graphs

Abstract

It is a long standing open problem to find an explicit description of the stable set polytope of clawfree graphs. Yet more than 20 years after the discovery of a polynomial algorithm for the maximum stable set problem for claw-free graphs, there is even no conjecture at hand today. Such a conjecture exists for the class of quasi-line graphs. This class of graphs is a proper superclass of line graphs and a proper subclass of claw-free graphs for which it is known that not all facets have 0/1 normal vectors. The Ben Rebea conjecture states that the stable set polytope of a quasi-line graph is completely described by clique-family inequalities. Chudnovsky and Seymour recently provided a decomposition result for claw-free graphs and proved that the Ben Rebea conjecture holds, if the quasi-line graph is not a fuzzy circular interval graph. In this paper, we give a proof of the Ben Rebea conjecture by showing that it also holds for fuzzy circular interval graphs. Our result builds upon an algorithm of Bartholdi, Orlin and Ratliff which is concerned with integer programs defined by circular ones matrices.

Coloring Fuzzy Circular Interval Graphs

“…If all fuzzy pairs are trivial, then the graph is a circular interval graph, see e.g. [3]. Lemma 1.1 Given a fuzzy circular interval graph G and a representation, if every fuzzy pair of G w.r.t.…”

Lovász-Schrijver PSD-Operator on Claw-Free Graphs

“…Based on the results of Eisenbrand et al [9] and Stauffer [33], combined with the characterization of LS + -imperfect webs from [11] (Theorem 4), we are able to show: Theorem 7. All facet-defining LS + -perfect fuzzy circular interval graphs are cliques, odd holes or odd antiholes.…”

“…Eisenbrand et al [9] proved that clique family inequalities suffice to describe the stable set polytope of fuzzy circular interval graphs. Stauffer [33] verified a conjecture of [27] that every facet-defining clique family inequality of a fuzzy circular interval graph G is associated with a web in G.…”

Vertex adjacencies in the set covering polyhedron

“…Finally, we set The previous corollary is also a consequence of a technique by Bartholdi et al [3], which Eisenbrand et al [8] employed to show that if A is row circular, then the slices {x ∈ Q(A) | 1 · x = β} are integral polytopes for β ∈ Z.…”