Weighted graphs defining facets: a connection between stable set and linear ordering polytopes
| dc.creator | Doignon, Jean-Paul | |
| dc.creator | Fiorini, Samuel | |
| dc.creator | Joret, Gwenaël | |
| dc.date | 2008-09-22 | |
| dc.date.accessioned | 2026-07-07T12:12:18Z | |
| dc.date.available | 2026-07-07T12:12:18Z | |
| dc.description | 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 Lovász (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, Chvátal (1975) shows that each alpha-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 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 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 alpha-critical graphs. | |
| dc.identifier | https://arxiv.org/abs/0809.3724 | |
| dc.identifier | http://arxiv.org/abs/0809.3724 | |
| dc.identifier | Discrete Optimization, Vol. 6 (1), 2009, pp. 1-9 | |
| dc.identifier | doi:10.1016/j.disopt.2008.07.001 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/210506 | |
| dc.subject | Combinatorics | |
| dc.title | Weighted graphs defining facets: a connection between stable set and linear ordering polytopes | |
| dc.type | text |