  A matching $M$ in a graph $G$ is uniquely restricted if there is no matching
$M'$ in $G$ that is distinct from $M$ but covers the same vertices as $M$.
Solving a problem posed by Golumbic, Hirst, and Lewenstein, we characterize the
graphs in which some maximum matching is uniquely restricted. Solving a problem
posed by Levit and Mandrescu, we characterize the graphs in which every maximum
matching is uniquely restricted. Both our characterizations lead to efficient
recognition algorithms for the corresponding graphs.
