  Let $G=(V,E)$ be a finite undirected graph. An edge subset $E' \subseteq E$
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 $G$. The
DIM problem is \NP-complete even for very restricted graph classes such as
planar bipartite graphs with maximum degree 3 but was solved in linear time for
$P_7$-free graphs and in polynomial time for $P_8$-free graphs. In this paper,
we solve it in polynomial time for $P_9$-free graphs.
