Weighted Efficient Domination for $P_6$-Free Graphs in Polynomial Time

In a finite undirected graph G = (V,E), a vertex v ∈ V dominates itself and its neighbors in G. A vertex set D ⊆ V is an efficient dominating set (e.d. for short) of G if every v ∈ V is dominated in G 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 for P7-free graphs but solvable in linear time for P5-free graphs. The P6-free case was the last open question for ED on F -free graphs. Very recently, Lokshtanov, Pilipczuk and van Leeuwen showed that weighted ED is solvable in polynomial time for P6-free graphs, based on their sub-exponential algorithm for the Maximum Weight Independent Set problem for P6-free graphs. Independently, at the same time, Mosca found a polynomial time algorithm for weighted ED on P6-free graphs using a direct approach. In this paper, we describe the details of this approach which is based on modular decomposition.