Graphs in which some and every maximum matching is uniquely restricted

TL;DR

This paper characterizes graphs where some or all maximum matchings are uniquely restricted, providing efficient algorithms to recognize such graphs, thus advancing understanding of matching uniqueness in graph theory.

Contribution

It offers the first complete characterizations of graphs with some or all maximum matchings uniquely restricted, solving open problems and enabling efficient recognition algorithms.

Findings

Characterization of graphs with some maximum matching uniquely restricted

Characterization of graphs with all maximum matchings uniquely restricted

Development of efficient recognition algorithms for these graph classes

Abstract

A matching $M$ in a graph $G$ is uniquely restricted if there is no matching $M'$ in $G$ that is distinct from $M$ but covers the same vertices as $M$. Solving a problem posed by Golumbic, Hirst, and Lewenstein, we characterize the graphs in which some maximum matching is uniquely restricted. Solving a problem posed by Levit and Mandrescu, we characterize the graphs in which every maximum matching is uniquely restricted. Both our characterizations lead to efficient recognition algorithms for the corresponding graphs.