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

Abstract

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 [L. Lovász, Some finite basis theorems on graph theory, in: A. Hajnal, V.T. Sós (Eds.), Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. II, in: Colloquia Mathematica Societatis János Bolyai, vol. 18, North-Holland, Amsterdam, 1978, pp. 717–729] 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 [V. Chvátal, On certain polytopes associated with graphs, Journal of Combinatorial Theory Ser. B 18 (1975) 138–154] 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 [L. Lipták, L. Lovász, Facets with fixed defect of the stable set polytope, Mathematical Programming 88 (Ser. A) (2000) 33–44; L. Lipták, L. Lovász, Critical facets of the stable set polytope, Combinatorica 21 (2001) 61–88] introduce critical facet-graphs (which again produce facet-defining inequalities for their stable set polytopes) and they establish a finite basis theorem. Second, Koppen [M. Koppen, Random utility representation of binary choice probabilities: Critical graphs yielding critical necessary conditions, Journal of Mathematical Psychology 39 (1995) 21–39] describes a construction that delivers from any α -critical graph a facet-defining inequality for the linear ordering polytope. Doignon et al. [J.-P. Doignon, S. Fiorini, G. Joret, Facets of the linear ordering polytope: A unification for the fence family through weighted graphs, Journal of Mathematical Psychology 50 (3) (2006) 251–262] 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 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 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.