A construction for non-rank facets of stable set polytopes of webs

Abstract

Graphs with circular symmetry, called webs, are relevant w.r.t. describing the stable set polytopes of two larger graph classes, quasi-line graphs [G. Giles, L.E. Trotter Jr., On stable set polyhedra for K 1 , 3 -free graphs, J. Combin. Theory B 31 (1981) 313–326; G. Oriolo, Clique family inequalities for the stable set polytope for quasi-line graphs, in: Stability Problems, Discrete Appl. Math. 132 (2003) 185–201 (special issue)] and claw-free graphs [A. Galluccio, A. Sassano, The rank facets of the stable set polytope for claw-free graphs, J. Combin. Theory B 69 (1997) 1–38; G. Giles, L.E. Trotter Jr., On stable set polyhedra for K 1 , 3 -free graphs, J. Combin. Theory B 31 (1981) 313–326]. Providing a decent linear description of the stable set polytopes of claw-free graphs is a long-standing problem [M. Grötschel, L. Lovász, A. Schrijver, Geometric Algorithms and Combinatorial Optimization, Springer-Verlag, 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 ≤ 3 have rank facets only [G. Dahl, Stable set polytopes for a class of circulant graphs, SIAM J. Optim. 9 (1999) 493–503; L.E. Trotter, Jr., A class of facet producing graphs for vertex packing polyhedra, Discrete Math. 12 (1975) 373–388] while there are examples with clique number ≥ 4 having non-rank facets [J. Kind, Mobilitätsmodelle für zellulare Mobilfunknetze: Produktformen und Blockierung, Ph.D. Thesis, RWTH Aachen, 2000; T.M. Liebling, G. Oriolo, B. Spille, G. Stauffer, On non-rank facets of the stable set polytope of claw-free graphs and circulant graphs, Math. Methods Oper. Res. 59 (2004) 25; G. Oriolo, Clique family inequalities for the stable set polytope for quasi-line graphs, in: Stability Problems, Discrete Appl. Math. 132 (2003) 185–201 (special issue); A. Pêcher, A. Wagler, On non-rank facets of stable set polytopes of webs with clique number four, Discrete Appl. Math. 154 (2006) 1408–1415]. In this paper, we provide a construction for non-rank facets of stable set polytopes of webs. This construction is the main tool to obtain in a companion paper [A. Pêcher, A. Wagler, Almost all webs are not rank-perfect, Math. Program 105 (2006) 311–328], for all fixed values of ω ≥ 5 that there are only finitely many webs with clique number ω whose stable set polytopes admit rank facets only.