Weighted Efficient Domination in Classes of P6-free Graphs

In a graph G, an efficient dominating set is a subset D of vertices such that D is an independent set and each vertex outside D has exactly one neighbor in D. The Minimum Weight Efficient Dominating Set (Min-WED) problem asks for an efficient dominating set of total minimum weight in a given vertex-weighted graph; the Maximum Weight Efficient Dominating Set (Max-WED) problem is defined similarly. The Min-WED/Max-WED is known to be NP -complete for P7-free graphs, and is known to be polynomial time solvable for P5-free graphs. However, the computational complexity of the min-WED/max-WED problem is unknown for P6free graphs. In this paper, we show that the Min-WED/Max-WED problem can be solved in polynomial time for two subclasses of P6free graphs, namely for (P6, S1,1,3)-free graphs, and for (P6, bull)-free graphs.