The Weighted Perfect Domination Problem and Its Variants

Abstract 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.

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