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

Abstract: In a finite undirected graph $G=(V,E)$, a vertex $v \in V$ {\em dominates} itself and its neighbors in $G$. A vertex set $D \subseteq V$ is an {\em efficient dominating set} ({\em e.d.} for short) of $G$ if every $v \in V$ is dominated in $G$ by exactly one vertex of $D$. The {\em Efficient Domination} (ED) problem, which asks for the existence of an e.d. in $G$, is known to be NP-complete for $P_7$-free graphs but solvable in polynomial time for $P_5$-free graphs. The $P_6$-free case was the last open question for the complexity of ED on $F$-free graphs. Recently, Lokshtanov, Pilipczuk and van Leeuwen showed that weighted ED is solvable in polynomial time for $P_6$-free graphs, based on their sub-exponential algorithm for the Maximum Weight Independent Set problem for $P_6$-free graphs. Independently, at the same time, Mosca found a polynomial time algorithm for weighted ED on $P_6$-free graphs using a direct approach. In this paper, we describe the details of this approach which is simpler and much faster, namely its time bound is ${\cal O}(n^6 m)$.