Equistable Graphs Research Articles

Short proofs on the structure of general partition, equistable and triangle graphs

While presenting a combinatorial point of view to the class of equistable graphs, Miklavič and Milanič pointed out the inclusions among the classes of equistable, general partition and triangle graphs. Orlin conjectured that the first two classes were equivalent, but in 2014, Milanič and Trotignon showed that the three classes are all distinct, leading to the following hierarchy: general partition graphs ⊂ equistable graphs ⊂ triangle graphs.In this paper, we solve an open problem from Anbeek et al. (1997) by showing that all the above classes are equivalent when restricted to planar graphs. Additionally, we present short proofs on some structural properties of the general partition, equistable and triangle classes. In particular, we show that the general partition and triangle classes are both closed under the operations of substitution, induction and contraction of modules which allow us to recover several results in the literature.

Equistarable Graphs and Counterexamples to Three Conjectures on Equistable Graphs

AbstractEquistable 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. Strongly equistable graphs are graphs such that for every and every nonempty subset T of vertices that is not a maximal stable set, there exist positive vertex weights assigning weight 1 to every maximal stable set such that the total weight of T does not equal c. 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. General partition graphs are exactly the graphs every edge of which is contained in a strong clique. In 1994, Mahadev, Peled, and Sun proved that every strongly equistable graph is equistable, and conjectured that the converse holds as well. 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, posed by Miklavič and Milanič in 2011, states that every equistable graph has a strong clique. 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. We also show that not all strongly equistable graphs are general partition.

Strong cliques and equistability of EPT graphs

In this paper, we characterize the equistable graphs within the class of EPT graphs, the edge-intersection graphs of paths in a tree. This result generalizes a previously known characterization of equistable line graphs. Our approach is based on the combinatorial features of triangle graphs and general partition graphs. We also show that, in EPT graphs, testing whether a given clique is strong is co-NP-complete. We obtain this hardness result by first showing hardness of the problem of determining whether a given graph has a maximal matching disjoint from a given edge cut. As a positive result, we prove that the problem of testing whether a given clique is strong is polynomial in the class of local EPT graphs, which are defined as the edge intersection graphs of paths in a star and are known to coincide with the line graphs of multigraphs.

Equistarable bipartite graphs

Recently, Milanič and Trotignon introduced the class of equistarable graphs as graphs without isolated vertices admitting positive weights on the edges such that a subset of edges is of total weight 1 if and only if it forms a maximal star. Based on equistarable graphs, counterexamples to three conjectures on equistable graphs were constructed, in particular to Orlin’s conjecture, which states that every equistable graph is a general partition graph.In this paper we characterize equistarable bipartite graphs. We show that a bipartite graph is equistarable if and only if every 2-matching of the graph extends to a matching covering all vertices of degree at least 2. As a consequence of this result, we obtain that Orlin’s conjecture holds within the class of complements of line graphs of bipartite graphs.We also connect equistarable graphs to the triangle condition, a combinatorial condition known to be necessary (but in general not sufficient) for equistability. We show that the triangle condition implies general partitionability for complements of line graphs of forests, and construct an infinite family of triangle non-equistable graphs within the class of complements of line graphs of bipartite graphs.

Equistable simplicial, very well-covered, and line graphs

We verify the conjectures of Mahadev–Peled–Sun and of Orlin, both related to equistable graphs, for the classes of simplicial, very well-covered and line graphs. Our results are based on the combinatorial features of triangle graphs and general partition graphs. In particular, we obtain several equivalent characterizations of equistable simplicial graphs, equistable very well-covered graphs, and equistable line graphs, some of which imply polynomial time recognition algorithms for graphs in these classes.

Equistable graphs, general partition graphs, triangle graphs, and graph products

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.

Complexity results for equistable graphs and related classes

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.

Equistable distance-hereditary graphs

A graph is called equistable when there is a non-negative weight function on its vertices such that a set S of vertices has total weight 1 if and only if S is maximal stable. We show that a necessary condition for a graph to be equistable is sufficient when the graph in question is distance-hereditary. This is used to design a polynomial-time recognition algorithm for equistable distance-hereditary graphs.

Equistable graphs

AbstractAn 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.

A class of threshold and domishold graphs: equistable and equidominating graphs

A threshold graph (respectively domishold graph) is a graph for which the independent sets (respectively the dominating sets) can be characterized by the 0, 1-solutions of a linear Inequality (see [1] and [3]). We define here the graphs for which the maximal independent sets (respectively the minimal dominating sets) are characterized by the 0, 1-solutions of a linear equation. Such graphs are said to be equistable (respectively equldominating). We characterize (by their architectural structure and by forbidden induced subgraphs) threshold graphs and domishold graphs which are equistable or equidominating. A larger class of equistable graphs is also presented.