Summary: 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, \textit{D. Lokshtanov} et al. [``Independence and efficient domination on \(P_6\)-free graphs'', in: Proceeding of SODA '16 Proceedings of the twenty-seventh annual ACM-SIAM symposium on Discrete algorithms. Arlington, Virginia: ACM. 1784--1803 (2016) ] 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 \({\mathcal O}(n^5 m)\) 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.