Weighted efficient domination problem on some perfect graphs Original Research Article

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 efficient dominating set of G if every vertex in V is dominated by exactly one vertex in D. The efficient domination problem is to find an efficient dominating set of G with minimum cardinality. Suppose that each vertex v∈V is associated with a weight. Then, the weighted efficient domination problem is to find an efficient dominating set with the minimum weight in G. In this paper, we show that the efficient 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 efficient domination problem on bipartite permutation graphs and distance-hereditary graphs.