  Let $G$ be a finite undirected graph with edge set $E$. An edge set $E'
\subseteq E$ is an {\em induced matching} in $G$ if the pairwise distance of
the edges of $E'$ in $G$ is at least two; $E'$ is {\em dominating} in $G$ if
every edge $e \in E \setminus E'$ intersects some edge in $E'$. The
\emph{Dominating Induced Matching Problem} (\emph{DIM}, for short) asks for the
existence of an induced matching $E'$ which is also dominating in $G$; this
problem is also known as the \emph{Efficient Edge Domination} Problem.
  The DIM problem is related to parallel resource allocation problems, encoding
theory and network routing. It is \NP-complete even for very restricted graph
classes such as planar bipartite graphs with maximum degree three. However, its
complexity was open for $P_k$-free graphs for any $k \ge 5$; $P_k$ denotes a
chordless path with $k$ vertices and $k-1$ edges. We show in this paper that
the weighted DIM problem is solvable in linear time for $P_7$-free graphs in a
robust way.
