  For a graph $G$ let $L(G)$ and $l(G)$ denote the size of the largest and
smallest maximum matching of a graph obtained from $G$ by removing a maximum
matching of $G$. We show that $L(G)\leq 2l(G),$ and $L(G)\leq (3/2)l(G)$
provided that $G$ contains a perfect matching. We also characterize the class
of graphs for which $L(G)=2l(G)$. Our characterization implies the existence of
a polynomial algorithm for testing the property $L(G)=2l(G)$. Finally we show
that it is $NP$-complete to test whether a graph $G$ containing a perfect
matching satisfies $L(G)=(3/2)l(G)$.
