  We consider the following question: given an $(X,Y)$-bigraph $G$ and a set $S
\subset X$, does $G$ contain two disjoint matchings $M_1$ and $M_2$ such that
$M_1$ saturates $X$ and $M_2$ saturates $S$? When $|S|\geq |X|-1$, this
question is solvable by finding an appropriate factor of the graph. In
contrast, we show that when $S$ is allowed to be an arbitrary subset of $X$,
the problem is NP-hard.
