On determining the imperfection ratio

Perfect graphs constitute a well-studied graph class with a rich structure, reflected by many characterizations w.r.t different concepts. Perfect graphs are, e.g., characterized as precisely those graphs G where the stable set polytope STAB(G) coincides with the clique constraint stable set polytope QSTAB(G). For all imperfect graphs STAB(G) ⊂ QSTAB(G) holds and, therefore, it is natural to measure imperfection in terms of the difference between STAB(G) and QSTAB(G). Several concepts have been developed in this direction, for instance the dilation ratio of STAB(G) and QSTAB(G) which is equivalent to the imperfection ratio imp(G) of G. To determine imp(G), both knowledge on the facets of STAB(G) and the extreme points of QSTAB(G) is required. For that, we extend a well-known result on antiblocking polyhedra by establishing a 1-1 correspondence between extreme points of QSTAB(G) and facet-defining subgraphs of G. We discuss several consequences, in particular, we give alternative proofs of several well-known results.