# Equistable graphs

**ABSTRACT** An equistable graph is a graph for which the incidence vectors of the maximal stable sets are the 0–1 solutions of a linear equation. A necessary condition and a sufficient condition for equistability are given. They are used to characterize the equistability of various classes of perfect graphs, outerplanar graphs, and pseudothreshold graphs. Some classes of equistable graphs are shown to be closed under graph substitution.

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**ABSTRACT:**A graph is called equistable when there is a nonnegativeweight function on its vertices such that a set S of verticeshas total weight 1 if and only if S is maximal stable. We show thata necessary condition for a graph to be equistable is su#cient whenthe graph in question is distance-hereditary. This is used to designa polynomial-time recognition algorithm for equistable distancehereditarygraphs.Discrete Applied Mathematics 02/2008; 156(4):462-477. DOI:10.1016/j.dam.2006.06.018 · 0.68 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**The class of equistable graphs is defined by the existence of a cost structure on the vertices such that the maximal stable sets are characterized by their costs. This graph class, not contained in any nontrivial hereditary class, has so far been studied mostly from a structural point of view; characterizations and polynomial time recognition algorithms have been obtained for special cases. We focus on complexity issues for equistable graphs and related classes. We describe a simple pseudo-polynomial-time dynamic programming algorithm to solve the maximum weight stable set problem along with the weighted independent domination problem in some classes of graphs, including equistable graphs. Our results are obtained within the wider context of Boolean optimization; corresponding hardness results are also provided. More specifically, we show that the above problems are APX-hard for equistable graphs and that it is co-NP-complete to determine whether a given cost function on the vertices of a graph defines an equistable cost structure of that graph.Annals of Operations Research 01/2011; 188(1):359-370. DOI:10.1007/s10479-010-0720-3 · 1.10 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**In this paper we examine the connections between equistable graphs, general partition graphs and triangle graphs. While every general partition graph is equistable and every equistable graph is a triangle graph, not every triangle graph is equistable, and a conjecture due to Jim Orlin states that every equistable graph is a general partition graph. The conjecture holds within the class of chordal graphs; if true in general, it would provide a combinatorial characterization of equistable graphs.Exploiting the combinatorial features of triangle graphs and general partition graphs, we verify Orlin’s conjecture for several graph classes, including AT-free graphs and various product graphs. More specifically, we obtain a complete characterization of the equistable graphs that are non-prime with respect to the Cartesian or the tensor product, and provide some necessary and sufficient conditions for the equistability of strong, lexicographic and deleted lexicographic products. We also show that the general partition graphs are not closed under the strong product, answering a question by McAvaney et al.Discrete Applied Mathematics 07/2011; 159(11):1148–1159. DOI:10.1016/j.dam.2011.03.011 · 0.68 Impact Factor