Dominating Induced Matchings for $P_8$-free Graphs in Polynomial Time

TL;DR

This paper proves that the weighted dominating induced matching problem can be solved efficiently in polynomial time for graphs that do not contain a chordless path of length 8, extending known results for shorter paths.

Contribution

It establishes the polynomial-time solvability of the weighted DIM problem specifically for $P_8$-free graphs, filling a gap in complexity classification.

Findings

Weighted DIM problem is polynomial-time solvable for $P_8$-free graphs.

Extends previous results from $P_7$-free graphs to $P_8$-free graphs.

Provides new algorithmic insights for a class of graph problems.

Abstract

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.