Determining All Maximum Uniquely Restricted Matching in Bipartite Graphs

TL;DR

This paper extends a digraph mapping approach to efficiently identify bipartite graphs where all maximum matchings are uniquely restricted, solving an open problem with polynomial-time algorithms.

Contribution

It introduces a method to recognize bipartite graphs with all maximum matchings uniquely restricted using acyclic digraph recognition, answering an open question.

Findings

Recognition of such bipartite graphs is polynomial-time solvable.

Mapping to acyclic digraphs is effective for the problem.

Provides a new approach for the forcing set problem in bipartite graphs.

Abstract

The approach mapping from a matching of bipartite graphs to digraphs has been successfully used for forcing set problem, in this paper, it is extended to uniquely restricted matching problem. We show to determine a uniquely restricted matching in a bipartite graph is equivalent to recognition a acyclic digraph. Based on these results, it proves that determine the bipartite graphs with all maximum matching are uniquely restricted is polynomial time. This answers an open question of Levit and Mandrescu(Discrete Applied Mathematics 132(2004) 163-164).