  In a graph $G$, an efficient dominating set is a subset $D$ of vertices such
that $D$ is an independent set and each vertex outside $D$ has exactly one
neighbor in $D$. The Minimum Weight Efficient Dominating Set (Min-WED) problem
asks for an efficient dominating set of total minimum weight in a given
vertex-weighted graph; the Maximum Weight Efficient Dominating Set (Max-WED)
problem is defined similarly. The Min-WED/Max-WED is known to be $NP$-complete
for $P_7$-free graphs, and is known to be polynomial time solvable for
$P_5$-free graphs. However, the computational complexity of the Min-WED/Max-WED
problem is unknown for $P_6$-free graphs. In this paper, we show that the
Min-WED/Max-WED problem can be solved in polynomial time for two subclasses of
$P_6$-free graphs, namely for ($P_6,S_{1,1,3}$)-free graphs, and for ($P_6$,
bull)-free graphs.
