  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.
