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.Facultad de Ciencias ExactasConsejo Nacional de Investigaciones Científicas y Técnica

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.Facultad de Ciencias ExactasConsejo Nacional de Investigaciones Científicas y Técnica

On the complexity of the independent set problem in triangle graphs

http://www.emse.fr/spip/IMG/pdf/RR_2009-500-006_abstract.pdfWe consider the complexity of the maximum independent set problem within triangle graphs, i.e., graphs G satisfying the following triangle condition: for every maximal independent set I in G and every edge uv in G¡I, there is a vertex w 2 I such that fu; v;wg is a triangle in G. We also introduce a new graph parameter (upper independent neighborhood number) and the corresponding upper independent neighborhood set problem. We show that for triangle graphs the new parameter equals to the independence number. Finally, we prove that the problems under consideration are NP-complete for triangle graphs and provide some polynomially solvable cases for these problems within triangle graphs

On the complexity of the independent set problem in triangle graphs

International audienceWe consider the complexity of the maximum (maximum weight) independent set problem within triangle graphs, i.e., graphs G satisfying the following triangle condition: for every maximal independent set I in G and every edge uv in G − I, there is a vertex w ∈ I such that {u, v,w} is a triangle in G. We also introduce a new graph parameter (the upper independent neighborhood number) and the corresponding upper independent neighborhood set problem. We show that for triangle graphs the new parameter is equal to the independence number. We prove that the problems under consideration are NPcomplete, even for some restricted subclasses of triangle graphs, and provide several polynomially solvable cases for these problems within triangle graphs. Furthermore, we show that, for general triangle graphs, the maximum independent set problem and the upper independent neighborhood set problem cannot be polynomially approximated within any fixed constant factor greater than one unless P = NP