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 Equidomination in Graphs

A graph G=(V,E) is called equidominating if there exists a value t in IN and a weight function w : V -> IN such that the total weight of a subset D of V is equal to t if and only if D is a minimal dominating set. Further, w is called an equidominating function, t a target value and the pair (w,t) an equidominating structure. To decide whether a given graph is equidominating is referred to as the EQUIDOMINATION problem. First, we examine several results on standard graph classes and operations with respect to equidomination. Furthermore, we characterize hereditarily equidominating graphs. These are the graphs whose every induced subgraph is equidominating. For those graphs, we give a finite forbidden induced subgraph characterization and a structural decomposition. Using this decomposition, we state a polynomial time algorithm that recognizes hereditarily equidominating graphs. We introduce two parameterized versions of the EQUIDOMINATION problem: the k-EQUIDOMINATION problem and the TARGET-t EQUIDOMINATION problem. For k in IN, a graph is called k-equidominating if we can identify the minimal dominating sets using only weights from 1 to k. In other words, if an equidominating function with co-domain {1,...,k} exists. For t in IN, a graph is said to be target-t equidominating if there is an equidominating structure with target value t. For both parameterized problems we prove fixed-parameter tractability. The first step for this is to achieve the so-called pseudo class partition, which coarsens the twin partition. It is founded on the requirement that vertices from different blocks of the partition cannot have equal weights in any equidominating structure. Based on the pseudo class partition, we state an XP algorithm for the parameterized versions of the EQUIDOMINATION problem. The second step is the examination of three reduction rules - each of them concerning a specific type of block of the pseudo class partition - which we use to construct problem kernels. The sizes of the kernels are bounded by a function depending only on the respective parameter. By applying the XP algorithm to the kernels, we achieve FPT algorithms. The concept of equidomination was introduced nearly 40 years ago, but hardly any investigations exist. With this thesis, we want to fill that gap. We may lay the foundation for further research on equidomination

Computing Well-Covered Vector Spaces of Graphs using Modular Decomposition

A graph is well-covered if all its maximal independent sets have the same cardinality. This well studied concept was introduced by Plummer in 1970 and naturally generalizes to the weighted case. Given a graph $G$, a real-valued vertex weight function $w$ is said to be a well-covered weighting of $G$ if all its maximal independent sets are of the same weight. The set of all well-covered weightings of a graph $G$ forms a vector space over the field of real numbers, called the well-covered vector space of $G$. Since the problem of recognizing well-covered graphs is $\mathsf{co}$-$\mathsf{NP}$-complete, the problem of computing the well-covered vector space of a given graph is $\mathsf{co}$-$\mathsf{NP}$-hard. Levit and Tankus showed in 2015 that the problem admits a polynomial-time algorithm in the class of claw-free graph. In this paper, we give two general reductions for the problem, one based on anti-neighborhoods and one based on modular decomposition, combined with Gaussian elimination. Building on these results, we develop a polynomial-time algorithm for computing the well-covered vector space of a given fork-free graph, generalizing the result of Levit and Tankus. Our approach implies that well-covered fork-free graphs can be recognized in polynomial time and also generalizes some known results on cographs.Comment: 25 page