  Let $G=(V,E)$ be a finite undirected graph. An edge set $E' \subseteq E$ is a
{\em dominating induced matching} ({\em d.i.m.}) in $G$ if every edge in $E$ is
intersected by exactly one edge of $E'$. The \emph{Dominating Induced Matching}
(\emph{DIM}) problem asks for the existence of a d.i.m.\ in $G$; this problem
is also known as the \emph{Efficient Edge Domination} problem; it is the
Efficient Domination problem for line graphs.
  The DIM problem is \NP-complete even for very restricted graph classes such
as planar bipartite graphs with maximum degree 3 but is solvable in linear time
for $P_7$-free graphs, and in polynomial time for $S_{1,2,4}$-free graphs as
well as for $S_{2,2,2}$-free graphs and for $S_{2,2,3}$-free graphs. In this
paper, combining two distinct approaches, we solve it in polynomial time for
$S_{1,1,5}$-free graphs.
