Almost all webs with odd clique number 5 are not rank - perfect

Graphs with circular symmetry, called webs, are relevant w.r.t. describing the stable set polytopes of two larger graph classes, quasi-line graphs [7,11] and claw-free graphs [6,7]. Providing a decent linear description of the stable set polytopes of claw-free graphs is a long-standing problem [8]. 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 [4,15] while there are examples with clique number 4 having non-rank facets [9–11,13]. In this paper we prove, with the help of a construction for non-rank facets [14], that there are only finitely many webs with odd clique number at least 5 whose stable set polytopes admit rank facets only.