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 ED 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. © 2002 Elsevier Science B.V.