Equistarable graphs and counterexamples to three conjectures on equistable graphs

Equistable graphs are graphs admitting positive weights on vertices such that a subset of vertices is a maximal stable set if and only if it is of total weight 1. In 1994, Mahadev et al. introduced a subclass of equistable graphs, called strongly equistable graphs, as graphs such that for every c ≤ 1 and every non-empty subset T of vertices that is not a maximal stable set, there exist positive vertex weights such that every maximal stable set is of total weight 1 and the total weight of T does not equal c. Mahadev et al. conjectured that every equistable graph is strongly equistable. General partition graphs are the intersection graphs of set systems over a finite ground set U such that every maximal stable set of the graph corresponds to a partition of U . In 2009, Orlin proved that every general partition graph is equistable, and conjectured that the converse holds as well. Orlin’s conjecture, if true, would imply the conjecture due to Mahadev, Peled, and Sun. An intermediate conjecture, one that would follow from Orlin’s conjecture and would imply the conjecture by Mahadev, Peled, and Sun, was posed by Miklavič and Milanič in 2011, and states that every equistable graph has a clique intersecting all maximal stable sets. The above conjectures have been verified for several graph classes. We introduce the notion of equistarable graphs and based on it construct counterexamples to all three conjectures within the class of complements of line graphs of triangle-free graphs. ∗M.M. is supported in part by “Agencija za raziskovalno dejavnost Republike Slovenije”, research program P1–0285 and research projects J1–4010, J1–4021, J1–5433, and N1–0011: GReGAS, supported in part by the European Science Foundation. N.T. is partially supported by ANR project Stint under reference ANR-13-BS02-0007. Part of this research was carried out during the visit of M.M. to N.T. at ENS Lyon; their support is gratefully acknowledged. 1 ar X iv :1 40 7. 16 70 v1 [ m at h. C O ] 7 J ul 2 01 4