# On the hardness of deciding the equality of the induced and the uniquely restricted matching number

## TL;DR

This paper proves that deciding whether the maximum induced matching equals the maximum uniquely restricted matching in a graph is NP-hard, indicating the problem's computational difficulty and the unlikelihood of a simple characterization.

## Contribution

It establishes the NP-hardness of the decision problem for equality of induced and uniquely restricted matching numbers in graphs.

## Key findings

- Deciding equality of the two matching numbers is NP-hard.
- The result suggests a likely absence of a simple characterization.
- The problem's complexity impacts related graph matching research.

## Abstract

If $G(M)$ denotes the subgraph of a graph $G$ induced by the set of vertices that are covered by some matching $M$ in $G$, then $M$ is an induced or a uniquely restricted matching if $G(M)$ is $1$-regular or if $M$ is the unique perfect matching of $G(M)$, respectively. Let $\nu_s(G)$ and $\nu_{ur}(G)$ denote the maximum cardinality of an induced and a uniquely restricted matching in $G$. Golumbic, Hirst, and Lewenstein (Uniquely restricted matchings, Algorithmica 31 (2001) 139-154) posed the problem to characterize the graphs $G$ with $\nu_{ur}(G) = \nu_{s}(G)$. We prove that the corresponding decision problem is NP-hard, which suggests that a good characterization is unlikely to be possible.