Clique family inequalities for the stable set polytope of quasi-line graphs

In one of fundamental work in combinatorial optimization Edmonds gave a complete linear description of the matching polytope. Matchings in a graph are equivalent to stable sets its line graph. Also the neighborhood of any vertex in a line graph partitions into two cliques: graphs with this latter property are called quasi-line graphs. Quasi-line graphs are a subclass of claw-free graphs, and as for claw-free graphs, there exists a polynomial algorithm for finding a maximum weighted stable set on such graphs, but we do not have a complete characterization of their stable set polytope (ssp). In the paper we introduce a class of inequalities, called clique-family inequalities, which are valid for the ssp of any graph and match the odd set inequalities defined by Edmonds for the matching polytope. This class of inequalities unifies all the known (non-trivial) facet inducing inequalities for the ssp of a quasi-line graph. We therefore conjecture that all the non-trivial facets of the ssp of a quasi-line graph belong to this class. We show that the conjecture is indeed correct for the classes of quasi-line graphs for which we have a complete description of the ssp. We discuss some approaches for solving the conjecture and a related problem.