Perfect edge domination and efficient edge domination in graphs

Abstract

Let G=( V, E) be a finite and undirected graph without loops and multiple edges. An edge is said to dominate itself and any edge adjacent to it. A subset D of E is called a perfect edge dominating set if every edge of E⧹ D is dominated by exactly one edge in D and an efficient edge dominating set if every edge of E is dominated by exactly one edge in D. The perfect (efficient) edge domination problem is to find a perfect (efficient) edge dominating set of minimum size in G. Suppose that each edge e is associated with a real number w( e) as its weight. Then, the weighted perfect (efficient) edge domination problem is to calculate a perfect (efficient) edge dominating set D such that the weight w( D) of D is minimum, where w( D)=∑ e∈ D w( e). In this paper, we show that the perfect (efficient) edge domination problem is NP-complete on bipartite (planar bipartite) graphs. Moreover, we present linear-time algorithms to solve the weighted perfect (efficient) edge domination problem on generalized series–parallel graphs and chordal graphs.