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

## TL;DR

This paper proves that the Dominating Induced Matching problem can be solved in polynomial time for $P_9$-free graphs, extending previous results for smaller path-free graph classes.

## Contribution

It extends the polynomial-time solvability of the DIM problem to $P_9$-free graphs, a previously unresolved class.

## Key findings

- DIM problem is polynomial-time solvable for $P_9$-free graphs.
- The complexity status of DIM is clarified for this class.
- Previous polynomial results for $P_7$ and $P_8$-free graphs are extended.

## Abstract

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.