On non-rank facets of stable set polytopes of webs with clique number four

Graphs with circular symmetry, called webs, are relevant for 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 = 4 having non-rank facets. The aim of the present paper is to treat the remaining case with clique number =4: we provide an infinite sequence of such webs whose stable set polytopes admit non-rank facets. s of claw-free graphs is a long-standing problem [M. Grotschel, L. Lovasz, A. Schrijver, Geometric Algorithms and Combinatorial Optimization, Springer, Berlin, 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 = 4 having non-rank facets [J. Kind, Mobilitatsmodelle fur zellulare Mobilfunknetze: Produktformen und Blockierung, Ph.D. Thesis, RWTH Aachen, 2000; G. Oriolo, Clique family inequalities for the stable set polytope for quasi-line graphs, Discrete Appl. Math. 132 (2004) 185-201; T. Liebling, G. Oriolo, B. Spille, G. Stauffer, On non-rank facets of the stable set polytope of claw-free graphs and circulant graphs, Math. Meth. Oper. Res. 59 (2004) 25-35]. The aim of the present paper is to treat the remaining case with clique number =4: we provide an infinite sequence of such webs whose stable set polytopes admit non-rank facets.

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