TY - GEN A1 - Pecher, Arnaud A1 - Wagler, Annegret T1 - On Non-Rank Facets of Stable Set Polytopes of Webs with Clique Number Four N2 - Graphs with circular symmetry, called webs, are relevant w.r.t. describing the stable set polytopes of two larger graph classes, quasi-line graphs and claw-free graphs. Providing a decent linear description of the stable set polytopes of claw-free graphs is a long-standing problem. However, even the problem of finding all facets of stable set polytopes of webs is open. So far, it is only known that stable set polytopes of webs with clique number $\leq 3$ have rank facets only while there are examples with clique number $>4$ having non-rank facets. T3 - ZIB-Report - 03-01 KW - web KW - rank-perfect graph KW - stable set polytope KW - (non-)rank facet Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7238 ER - TY - GEN A1 - PĂȘcher, Arnaud A1 - Wagler, Annegret T1 - A construction for non-rank facets of stable set polytopes of webs N2 - Graphs with circular symmetry, called webs, are relevant for describing the stable set polytopes of two larger graph classes, quasi-line graphs [{\sl Giles and Trotter 1981, Oriolo 2001}] and claw-free graphs [{\sl Galluccio and Sassano 1997, Giles and Trotter 1981}]. Providing a decent linear description of the stable set polytopes of claw-free graphs is a long-standing problem [{\sl Grötschel, Lov\'asz, and Schrijver 1988}]. However, even the problem of finding all facets of stable set polytopes of webs is open. So far, it is only known that stable set polytopes of webs with clique number $\leq 3$ have rank facets only [{\sl Dahl 1999, Trotter 1975}] while there are examples with clique number $\geq 4$ having non-rank facets [{\sl e.g. Liebling et al. 2003, Oriolo 2001, P\^echer and Wagler 2003}]. In this paper, we provide a construction for non-rank facets of stable set polytopes of webs. We use this construction to prove, for several fixed values of $\omega$ including all odd values at least 5, that there are only finitely many webs with clique number $\omega$ whose stable set polytopes admit rank facets only. T3 - ZIB-Report - 03-21 KW - web KW - rank-perfect graph KW - stable set polytope KW - (non-)rank facet Y1 - 2003 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-7437 ER - TY - GEN A1 - Wagler, Annegret T1 - Antiwebs are Rank-Perfect N2 - \We discuss a nested collection of three superclasses of perfect graphs: near-perfect, rank-perfect, and weakly rank-perfect graphs. For that, we start with the description of the stable set polytope for perfect graphs and allow stepwise more general facets for the stable set polytopes of the graphs in each superclass. Membership in those three classes indicates how far a graph is away from being perfect. We investigate for webs and antiwebs to which of the three classes they belong. We provide a complete description of the facets of the stable set polytope for antiwebs (with help of a result due to Shepherd on near-bipartite graphs). The main result is that antiwebs are rankperfect. T3 - ZIB-Report - 02-07 KW - relaxations of perfect graphs KW - antiwebs KW - stable set polytope Y1 - 2002 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6742 ER - TY - GEN A1 - Koster, Arie M.C.A. A1 - Wagler, Annegret T1 - The extreme points of QSTAB(G) and its implications N2 - Perfect graphs constitute a well-studied graph class with a rich structure, reflected by many characterizations w.r.t different concepts. Perfect graphs are, e.g., characterized as precisely those graphs $G$ where the stable set polytope STAB$(G)$ coincides with the clique constraint stable set polytope QSTAB$(G)$. For all imperfect graphs STAB$(G) \subset$ QSTAB$(G)$ holds and, therefore, it is natural to measure imperfection in terms of the difference between STAB$(G)$ and QSTAB$(G)$. Several concepts have been developed in this direction, for instance the dilation ratio of STAB$(G)$ and QSTAB$(G)$ which is equivalent to the imperfection ratio imp$(G)$ of $G$. To determine imp$(G)$, both knowledge on the facets of STAB$(G)$ and the extreme points of QSTAB$(G)$ is required. The anti-blocking theory of polyhedra yields all {\em dominating} extreme points of QSTAB$(G)$, provided a complete description of the facets of STAB$(\overline G)$ is known. As this is typically not the case, we extend the result on anti-blocking polyhedra to a {\em complete} characterization of the extreme points of QSTAB$(G)$ by establishing a 1-1 correspondence to the facet-defining subgraphs of $\overline G$. We discuss several consequences, in particular, we give alternative proofs of several famous results. T3 - ZIB-Report - 06-30 KW - perfect graphs KW - imperfection ratio KW - stable set polytope Y1 - 2006 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-9249 ER -