Weighted efficient domination problem on some perfect graphs

Abstract 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.

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