14 research outputs found

    Equistarable graphs and counterexamples to three conjectures on equistable graphs

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    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 11. In 19941994, Mahadev et al.~introduced a subclass of equistable graphs, called strongly equistable graphs, as graphs such that for every c1c \le 1 and every non-empty subset TT of vertices that is not a maximal stable set, there exist positive vertex weights such that every maximal stable set is of total weight 11 and the total weight of TT does not equal cc. 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 UU such that every maximal stable set of the graph corresponds to a partition of UU. In 20092009, 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 Miklavi\v{c} and Milani\v{c} in 20112011, 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

    Strong cliques and equistability of EPT graphs

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    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.Facultad de Ciencias ExactasConsejo Nacional de Investigaciones Científicas y Técnica

    Strong cliques and equistability of EPT graphs

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    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.Facultad de Ciencias ExactasConsejo Nacional de Investigaciones Científicas y Técnica

    Decomposing 1-Sperner hypergraphs

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    A hypergraph is Sperner if no hyperedge contains another one. A Sperner hypergraph is equilizable (resp., threshold) if the characteristic vectors of its hyperedges are the (minimal) binary solutions to a linear equation (resp., inequality) with positive coefficients. These combinatorial notions have many applications and are motivated by the theory of Boolean functions and integer programming. We introduce in this paper the class of 11-Sperner hypergraphs, defined by the property that for every two hyperedges the smallest of their two set differences is of size one. We characterize this class of Sperner hypergraphs by a decomposition theorem and derive several consequences from it. In particular, we obtain bounds on the size of 11-Sperner hypergraphs and their transversal hypergraphs, show that the characteristic vectors of the hyperedges are linearly independent over the reals, and prove that 11-Sperner hypergraphs are both threshold and equilizable. The study of 11-Sperner hypergraphs is motivated also by their applications in graph theory, which we present in a companion paper

    ХАРАКТЕРИЗАЦИЯ 1-ТРЕУГОЛЬНЫХ ГРАФОВ

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    A graph is called 1-triangle if for each maximal independent set I, each edge of this graph with both end vertices not belonging to I forms exactly one triangle with a vertex from the set I. We have obtained a structural characterization of 1-triangle graphs which implies a polynomial time recognition algorithm for this class of graphs.Граф называется 1-треугольным, если для любого максимального независимого множества I этого графа каждое ребро графа, не инцидентное ни одной вершине из I, образует единственный треугольник с вершиной из множества I. В работе получена структурная характеризация класса 1-треугольных графов, которая влечёт полиномиальный алгоритм их распознавания

    Recent results and open problems on CIS Graphs

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