Weighted efficient domination problem on some perfect graphs

Given a simple graph G = (V; E), a vertex v ∈ V is said to dominate itself and all vertices adjacent to it. A subset D of V is called an e cient dominating set of G if every vertex in V is dominated by exactly one vertex in D. The e cient domination problem is to 3nd an e cient dominating set of G with minimum cardinality. Suppose that each vertex v ∈ V is associated with a weight. Then, the weighted e cient domination problem is to 3nd an e cient dominating set with the minimum weight in G. In this paper, we show that the e cient domination problem is NP-complete for planar bipartite graphs and chordal bipartite graphs. Assume that a permutation diagram of a bipartite permutation graph and a one-vertex-extension ordering of a distance-hereditary graph are given in advance. Then, we give O(|V |) time algorithms for the weighted e cient domination problem on bipartite permutation graphs and distance-hereditary graphs. ? 2002 Elsevier Science B.V. All rights reserved.