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

A graph is alpha-critical if its stability number increases whenever an edge is removed from its edge set. The class of alpha-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 Lov\'asz (1978) is the finite basis theorem for alpha-critical graphs of a fixed defect. The class of alpha-critical graphs is also of interest for at least two topics of polyhedral studies. First, Chv\'atal (1975) shows that each alpha-critical graph induces a rank inequality which is facet-defining for its stable set polytope. Investigating a weighted generalization, Lipt\'ak and Lov\'asz (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 alpha-critical 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 alpha-critical graphs. We show that facet-defining graphs (for the linear ordering polytope) are obtainable from 1-critical 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 Lipt\'ak and Lov\'asz's finite basis theorem for critical facet-graphs. At the end of the paper we offer an alternative proof of Lov\'asz's finite basis theorem for alpha-critical graphs

The biorder polytope

Biorders, also called Ferrers relations, formalize Guttman scales. Irreflexive biorders on a set are exactly the interval orders on that set. The biorder polytope is the convex hull of the characteristic matrices of biorders. Its definition is thus similar to the definition of other order polytopes, the linear ordering polytope being the proeminent example. We investigate the combinatorial structure of the biorder polytope, thus obtaining a complete linear description in a specific case, and the automorphism group in all cases. Moreover, a class of facet-defining inequalities defined from weighted graphs is thoroughly analyzed. A weighted generalization of stability-critical graphs is presented, which leads to new facets even for the well-studied linear ordering polytope.SCOPUS: ar.jinfo:eu-repo/semantics/publishe

Facets of the linear ordering polytope: a unification for the fence family

The binary choice polytope appeared in the investigation of the binary choice problem formulated by Guilbaud and Block and Marschak. It is nowadays known to be the same as the linear ordering polytope from operations research (as studied by Grötschel, Jünger and Reinelt). The central problem is to find facet-defining linear inequalities for the polytope. Fence inequalities constitute a prominent class of such inequalities (Cohen and Falmagne; Grötschel, Jünger and Reinelt). Two different generalizations exist for this class: the reinforced fence inequalities of Leung and Lee, and independently Suck, and the stability-critical fence inequalities of Koppen. Together with the fence inequalities, these inequalities form the fence family. Building on previous work on the biorder polytope by Christophe, Doignon and Fiorini, we provide a new class of inequalities which unifies all inequalities from the fence family. The proof is based on a projection of polytopes. The new class of facet-defining inequalities is related to a specific class of weighted graphs, whose definition relies on a curious extension of the stability number. We investigate this class of weighted graphs which generalize the stability-critical graphs. © 2006 Elsevier Inc. All rights reserved.info:eu-repo/semantics/publishe