On some Graphs with a Unique Perfect Matching

TL;DR

This paper presents linear-time algorithms for detecting and finding unique perfect matchings in specific graph classes, and characterizes claw-free graphs with this property.

Contribution

It introduces efficient algorithms for unique perfect matching detection in cographs, split graphs, interval graphs, and claw-free graphs, along with a characterization for claw-free graphs.

Findings

Linear-time algorithms for specific graph classes

Constructive characterization of claw-free graphs with a unique perfect matching

Efficient detection and construction methods

Abstract

We show that deciding whether a given graph $G$ of size $m$ has a unique perfect matching as well as finding that matching, if it exists, can be done in time $O(m)$ if $G$ is either a cograph, or a split graph, or an interval graph, or claw-free. Furthermore, we provide a constructive characterization of the claw-free graphs with a unique perfect matching.