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1 LaBRI - Laboratoire Bordelais de Recherche en Informatique 2 Realopt - Reformulations based algorithms for Combinatorial Optimization LaBRI - Laboratoire Bordelais de Recherche en Informatique, IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest 3 IMO - Institute for Mathematical Optimization

Abstract : Graphs with circular symmetry, called webs, are crucial for describing the stable set polytopes of two larger graph classes, quasi-line graphs8,12 and claw-free graphs 7,8. Providing a complete 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 = 4 having non-rank facets 10 12,15. In this paper we prove, building on a construction for non-rank facets from 16, 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





Author: Arnaud Pecher - Annegret K. Wagler -

Source: https://hal.archives-ouvertes.fr/



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