  Let $G=(V,E)$ be a finite undirected graph without loops and multiple edges.
A subset $M \subseteq E$ of edges is a {\em dominating induced matching} ({\em
d.i.m.}) in $G$ if every edge in $E$ is intersected by exactly one edge of $M$.
In particular, this means that $M$ is an induced matching, and every edge not
in $M$ shares exactly one vertex with an edge in $M$. Clearly, not every graph
has a d.i.m.
  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 {\em Efficient Domination}
problem for line graphs.
  The DIM problem is \NP-complete in general, and even for very restricted
graph classes such as planar bipartite graphs with maximum degree 3. However,
DIM is solvable in polynomial time for claw-free (i.e., $S_{1,1,1}$-free)
graphs, for $S_{1,2,3}$-free graphs as well as for $S_{2,2,2}$-free graphs, in
linear time for $P_7$-free graphs, and in polynomial time for $P_8$-free graphs
($P_k$ is a special case of $S_{i,j,\ell}$). In a paper by Hertz, Lozin, Ries,
Zamaraev and de Werra, it was conjectured that DIM is solvable in polynomial
time for $S_{i,j,k}$-free graphs for every fixed $i,j,k$.
  In this paper, combining two distinct approaches, we solve it in polynomial
time for $S_{1,2,4}$-free graphs which generalizes the $S_{1,2,3}$-free as well
as the $P_7$-free case.
