Perfect edge domination: hard and solvable cases

Let G be an undirected graph. An edge of G dominates 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.