$\mathcal{P}$-matchings Parameterized by Treewidth

TL;DR

This paper studies the computational complexity of finding special matchings in graphs, where the subgraph induced by the matching’s endpoints satisfies a property, focusing on how this complexity relates to the graph's treewidth.

Contribution

It provides a parameterized complexity analysis of $ ext{P}$-matchings based on treewidth, offering insights into the tractability of these problems.

Findings

Complexity results for various $ ext{P}$-matchings with respect to treewidth.

Identification of fixed-parameter tractability or hardness for specific properties.

Framework for analyzing induced matchings and related problems in parameterized complexity.

Abstract

A \emph{matching} is a subset of edges in a graph $G$ that do not share an endpoint. A matching $M$ is a \emph{$\mathcal{P}$-matching} if the subgraph of $G$ induced by the endpoints of the edges of $M$ satisfies property $\mathcal{P}$. For example, if the property $\mathcal{P}$ is that of being a matching, being acyclic, or being disconnected, then we obtain an \emph{induced matching}, an \emph{acyclic matching}, and a \emph{disconnected matching}, respectively. In this paper, we analyze the problems of the computation of these matchings from the viewpoint of Parameterized Complexity with respect to the parameter \emph{treewidth}.