Solving the weighted efficient edge domination problem on bipartite permutation graphs Original Research Article

Given a simple graph G = (V,E), an edge (u, v)ϵ E is said to dominate itself and any edge (u,x) or (v,x), where xϵ V. A subset D⊂-E is called an efficient edge dominating set of G if all edges in E are dominated by exactly one edge of D. The efficient edge domination problem is to find an efficient edge dominating set of minimum size in G. Suppose that each edge eϵ E is associated with a real number w(e), called the weight of e. The weighted efficient edge domination problem is to calculate an efficient edge dominating set D of G such that the weight w(D) of D is minimum, where w(D) = ∑{w(e)¦eϵD}. In this paper, we show that the problem of determining whether G has an efficient edge dominating set is NP-complete when G is restricted to a bipartite graph. Consequently, the decision problem of efficient (vertex) domination remains NP-complete for the line graphs of bipartite graphs. Moreover, we present a linear time algorithm to solve the weighted efficient edge domination problem on bipartite permutation graphs, which form a subclass of bipartite graphs, using the technique of dynamic programming.