Weighted Efficient Domination for P5-Free Graphs in Linear Time

In a finite undirected graph G = (V,E), a vertex v ∈ V dominates itself and its neighbors. A vertex set D ⊆ V in G 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 for P7-free graphs but solvable in polynomial time for P5-free graphs. Very recently, it has been shown by Lokshtanov et al. and independently by Mosca that ED is solvable in polynomial time for P6-free graphs. In this note, we show that, based on modular decomposition, ED is solvable in linear time for P5-free graphs.