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 $d \geq 3$ 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 $P_5$-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 $P_5$.