The weighted perfect domination problem and its variants Original Research Article

A perfect dominating set of a graph G = (V, E) is a subset D of V such that every vertex not in D is adjacent to exactly one vertex in D. The perfect domination problem is to find the minimum size of a perfect dominating set of a graph. Suppose moreover that every vertex v ϵ V has a cost c(v) and every edge eϵE has a cost c(e). The weighted perfect domination problem is to find a perfect dominating set D such that its total cost c(D) = ∑{c(v): ϵD} + ∑{c(u, v): u∉D, vϵD and (u, v) ϵ E} is minimum. We also consider the following three variants of perfect domination. A perfect dominating set. D is called independent (resp. connected, total) if the subgraph 〈D〉 induced by D has no edge (resp. is connected, has no isolated vertex). This paper first proves that the three variants of perfect domination are NP-complete for bipartite graphs and chordal graphs, except for the connected perfect domination in chordal graphs. We then present linear-time algorithms for the weighted perfect domination problem and its three variants in block graphs.