Perfect edge domination : hard and solvable cases

Let G be an undirected graph. An edge of Gdominates itself and all edges adjacent to it. A subset E′ of edges of G is an edge dominating set of G, if every edge of the graph is dominated by some edge of E′. We say that E′ is a perfect edge dominating set of G, if every edge not in E′ is dominated by exactly one edge of E′. The perfect edge dominating problem is to determine a least cardinality perfect edge dominating set of G. For this problem, we describe two NP-completeness proofs, for the classes of claw-free graphs of degree at most 3, and for bounded degree graphs, of maximum degree at most d≥ 3 and large girth. In contrast, we prove that the problem admits an O(n) time solution, for cubic claw-free graphs. In addition, we prove a complexity dichotomy theorem for the perfect edge domination problem, based on the results described in the paper. Finally, we describe a linear time algorithm for finding a minimum weight perfect edge dominating set of a P5-free graph. The algorithm is robust, in the sense that, given an arbitrary graph G, either it computes a minimum weight perfect edge dominating set of G, or it exhibits an induced subgraph of G, isomorphic to a P5.Fil: Lin, Min Chih. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Cálculo; ArgentinaFil: Lozin, Vadim. University of Warwick; Reino UnidoFil: Moyano, Verónica Andrea. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Cálculo; ArgentinaFil: Szwarcfiter, Jayme L.. Universidade Federal do Rio de Janeiro; Brasil. Instituto Nacional de Metrologia, Qualidade e Tecnologia; Brasi

Dominating induced matchings in graphs containing no long claw

An induced matching in a graph is dominating if every edge not in shares exactly one vertex with an edge in . The DOMINATING INDUCED MATCHING problem (also known as EFFICIENT EDGE DOMINATION) asks whether a graph contains a dominating induced matching. This problem is generally NP-complete, but polynomial-time solvable for graphs with some special properties. In particular, it is solvable in polynomial time for claw-free graphs. In the present article, we provide a polynomial-time algorithmto solve the DOMINATING INDUCED MATCHING problem for graphs containing no long claw, that is, no induced subgraph obtained from the claw by subdividing each of its edges exactly once