Clique-circulants and the stable set polytope of fuzzy circular interval graphs

Abstract

In this paper, we give a complete and explicit description of the rank facets of the stable set polytope for a class of claw-free graphs, recently introduced by Chudnovsky and Seymour (Proceedings of the Bristish Combinatorial Conference, 2005), called fuzzy circular interval graphs. The result builds upon the characterization of minimal rank facets for claw-free graphs by Galluccio and Sassano (J. Combinatorial Theory 69:1–38, 2005) and upon the introduction of a superclass of circulant graphs that are called clique-circulants. The new class of graphs is invetigated in depth. We characterize which clique-circulants C are facet producing, i.e. are such that $$\sum_{v\in V(C)} x_v\leq\alpha(C)$$ is a facet of STAB(C), thus extending a result of Trotter (Discrete Math. 12:373–388, 1975) for circulants. We show that a simple clique family inequality (Oriolo, Discrete Appl. Math. 132(2):185–201, 2004) may be associated with each clique-circulant $$C\subseteq G$$, when G is fuzzy circular interval. We show that these inequalities provide all the rank facets of STAB(G), if G is a fuzzy circular interval graph. Moreover we conjecture that, in this case, they also provide all the non-rank facets of STAB(G) and offer evidences for this conjecture.