Efficient domination for classes of [formula omitted]-free graphs

Abstract

In a finite undirected graph G, a vertex dominates itself and all its neighbors in G. A vertex set D is an efficient dominating set (e.d. for short) of G if every vertex of G is dominated by exactly one vertex of D. The Efficient Domination (ED) problem, which asks for the existence of an e.d. in G, is known to be NP-complete even for very restricted graph classes such as P7-free chordal graphs; its complexity was an open question for P6-free graphs and was open even for the subclass of P6-free chordal graphs. Recently, Lokshtanov et al., and independently, Brandstädt and Mosca showed that ED is solvable in polynomial time for P6-free graphs.The ED problem on a graph G can be reduced to the Maximum Weight Independent Set problem on the square of G. In this paper, we show that squares of P6-free chordal graphs that have an e.d. are chordal; this even holds for the larger class of (P6, house, hole, domino)-free graphs. Thus, ED/WeightedED is solvable in polynomial time for (P6, house, hole, domino)-free graphs, and in particular, for P6-free chordal graphs. Also, the time bound achieved for ED on this class is much better than in the P6-free case.Moreover, we show that squares of P6-free graphs that have an e.d. are hole-free. Based on this result, we show that ED is solvable in polynomial time for (P6, net)-free graphs; again, the time bound for ED on this class is much better than in the P6-free case.