Weighted graphs defining facets: A connection between stable set and linear ordering polytopes

A graph is α-critical if its stability number increases whenever an edge is removed from its edge set. The class of α-critical graphs has several nice structural properties, most of them related to their defect which is the number of vertices minus two times the stability number. In particular, a remarkable result of Lovász (1978) is the finite basis theorem for α-critical graphs of a fixed defect. The class of α-critical graphs is also of interest for at least two topics of polyhedral studies. First, Chvátal (1975) shows that each α-critical graph induces a rank inequality which is facet-defining for its stable set polytope. Investigating a weighted generalization, Lipták and Lovász (2000, 2001) introduce critical facet-graphs (which again produce facet-defining inequalities for their stable set polytopes) and they establish a finite basis theorem. Second, Koppen (1995) describes a construction that delivers from any αcritical graph a facet-defining inequality for the linear ordering polytope. Doignon, Fiorini and Joret (2006) handle the weighted case and thus define facet-defining graphs. Here we investigate relationships between the two weighted generalizations of α-critical graphs. We show that facet-defining graphs (for the linear ordering polytope) are obtainable from 1critical facet-graphs (linked with stable set polytopes). We then use this connection to derive various results on facet-defining graphs, the most prominent one being derived from Lipták and Lovász’s finite basis theorem for critical facet-graphs. At the end of the paper we offer an alternative proof of Lovász’s finite basis theorem for α-critical graphs.