  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 ${\cal O}(n^6 m)$.
