Almost all webs 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 [8,12] and claw-free graphs [7,8]. Providing a decent linear description of the stable set polytopes of claw-free graphs is a long-standing problem [9]. Ben Rebea conjectured a description for quasi-line graphs, see [12]; Chudnovsky and Seymour [2] verified this conjecture recently for quasi-line graphs not belonging to the subclass of fuzzy circular interval graphs and showed that rank facets are required in this case only. Fuzzy circular interval graphs contain all webs and even the problem of finding all facets of their stable set polytopes is open. So far, it is only known that stable set polytopes of webs with clique number ≤ 3 have rank facets only [5,16] while there are examples with clique number≥ 4 having non-rank facets [10–12,14]. In this paper we prove, building on a construction for non-rank facets from [15], that the stable set polytopes of almost all webs with clique number≥ 5 admit non-rank facets. This adds support to the belief that these graphs are indeed the core of Ben Rebea’s conjecture. Finally, we present a conjecture how to construct all facets of the stable set polytopes of webs.