# Finding Dominating Induced Matchings in $S_{1,1,5}$-Free Graphs in Polynomial Time

## TL;DR

This paper proves that the Dominating Induced Matching problem can be solved in polynomial time for $S_{1,1,5}$-free graphs, extending the class of graphs where this problem is efficiently solvable.

## Contribution

The paper introduces a novel approach to solve the DIM problem in $S_{1,1,5}$-free graphs, expanding the known classes of graphs with polynomial-time solutions.

## Key findings

- DIM problem is polynomial-time solvable for $S_{1,1,5}$-free graphs.
- Combines two approaches to achieve the polynomial-time result.
- Extends the class of graphs with efficiently solvable DIM problem.

## Abstract

Let $G=(V,E)$ be a finite undirected graph. An edge set $E' \subseteq E$ 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 $G$; this problem is also known as the \emph{Efficient Edge Domination} problem; it is the Efficient Domination problem for line graphs. The DIM problem is \NP-complete even for very restricted graph classes such as planar bipartite graphs with maximum degree 3 but is solvable in linear time for $P_7$-free graphs, and in polynomial time for $S_{1,2,4}$-free graphs as well as for $S_{2,2,2}$-free graphs and for $S_{2,2,3}$-free graphs. In this paper, combining two distinct approaches, we solve it in polynomial time for $S_{1,1,5}$-free graphs.