On some Graphs with a Unique Perfect Matching
S. Chaplick, M. F\"urst, F. Maffray, D. Rautenbach

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 of size has a unique perfect matching as well as finding that matching, if it exists, can be done in time if 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.
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Taxonomy
TopicsAdvanced Graph Theory Research · Limits and Structures in Graph Theory · Graph Labeling and Dimension Problems
