  Let $G=(V,E)$ be a finite undirected graph. An edge set $E' \subseteq E$ is a
dominating induced matching (d.i.m.) in $G$ if every edge in $E$ is intersected
by exactly one edge of $E'$. The Dominating Induced Matching (DIM) problem asks
for the existence of a d.i.m. in $G$; this problem is also known as the
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 and is
solvable in linear time for $P_7$-free graphs. However, its complexity was open
for $P_k$-free graphs for any $k \ge 8$; $P_k$ denotes the chordless path with
$k$ vertices and $k-1$ edges. We show in this paper that the weighted DIM
problem is solvable in polynomial time for $P_8$-free graphs.
