Finding Dominating Induced Matchings in $P_9$-Free Graphs in Polynomial Time

Let $G=(V,E)$ be a finite undirected graph. An edge subset $E^{\mathrm{\prime }}\subseteq E$ is a {\em dominating induced matching} ({\em d.i.m.}) in $G$ if every edge in $E$ is intersected by exactly one edge of $E^{\mathrm{\prime }}$. 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.