On the Lovász–Schrijver PSD-operator on graph classes defined by clique cutsets

This work is devoted to the study of the Lovasz–Schrijver PSD-operator L S + applied to the edge relaxation ESTAB ( G ) of the stable set polytope STAB ( G ) of a graph G . In order to characterize the graphs G for which STAB ( G ) is achieved in one iteration of the L S + -operator, called L S + -perfect graphs, an according conjecture has been recently formulated ( L S + -Perfect Graph Conjecture). Here we study two graph classes defined by clique cutsets (pseudothreshold graphs and graphs without certain Truemper configurations). We completely describe the facets of the stable set polytope for such graphs, which enables us to show that one class is a subclass of L S + -perfect graphs, and to verify the L S + -Perfect Graph Conjecture for the other class.