Weighted Efficient Domination for $P_5$-Free and $P_6$-Free Graphs
Andreas Brandstädt, Raffaele Mosca · SIAM Journal on Discrete Mathematics · 2016
In a finite undirected graph $G=(V,E)$, a vertex $v \in V$ dominates itself and its neighbors in $G$. A vertex set $D \subseteq V$ is an efficient dominating set (e.d.s. for short) of $G$ if every $v \in V$ is dominated in $G$ by exactly one vertex of $D$. The Efficient Domination (ED) problem, which asks for the existence of an e.d.s. in $G$, is known to be NP-complete for $P_7$-free graphs and 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 quasi-polynomial algorithm for the Maximum Weight Independent Set problem for $P_6$-free graphs. Independently, by a direct approach which is simpler and faster, we found an ${\cal O}(n^5 m)$ time solution for weighted ED on $P_6$-free graphs. Moreover, we show that weighted ED is solvable in linear time for $P_5$-free graphs which solves another open question for the complexity of (weighted) ED. The result for $P_5$-free graphs is based on modular decomposition.