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

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.