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For a matching $M$ in a graph $G$, let $G(M)$ be the subgraph of $G$ inducedby the vertices of $G$ that are incident with an edge in $M$. The matching $M$is induced, if $G(M)$ is $1$-regular, and $M$ is uniquely restricted, if $M$ isthe unique perfect matching of $G(M)$. The induced matching number $\nu_s(G)$of $G$ is the largest size of an induced matching in $G$, and the uniquelyrestricted matching number $\nu_{ur}(G)$ of $G$ is the largest size of auniquely restricted matching in $G$.  Golumbic, Hirst, and Lewenstein (Uniquely restricted matchings, Algorithmica31 (2001) 139-154) posed the problem to characterize the graphs $G$ with$\nu_s(G)=\nu_{ur}(G)$. We give a complete characterization of the$2$-connected subcubic graphs $G$ of sufficiently large order with$\nu_s(G)=\nu_{ur}(G)$. As a consequence, we are able to show that the subcubicgraphs $G$ with $\nu_s(G)=\nu_{ur}(G)$ can be recognized in polynomial time.

On the equality of the induced matching number and the uniquely restricted matching number for subcubic graphs