743 eng reportzib 0 2003-07-03 2003-07-03 -- 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. 03-21 744 urn:nbn:de:0297-zib-7437 Appeared in: European Journal of Combinatorics 27 (2006) 1172-1185 Arnaud Pêcher Annegret Wagler 03-21 eng uncontrolled web eng uncontrolled rank-perfect graph eng uncontrolled stable set polytope eng uncontrolled (non-)rank facet Informatik, Informationswissenschaft, allgemeine Werke Perfect graphs Dominating sets, independent sets, cliques Polyhedral combinatorics, branch-and-bound, branch-and-cut ZIB Allgemein StableSets https://opus4.kobv.de/opus4-zib/files/743/ZR-03-21.ps https://opus4.kobv.de/opus4-zib/files/743/ZR-03-21.pdf