Three results towards approximation of special maximum matchings in graphs

For a graph $G$ define the parameters $\ell \left(G\right)$ and $L\left(G\right)$ as the minimum and maximum value of $\nu (G\setminus F)$, where $F$ is a maximum matching of $G$ and $\nu \left(G\right)$ is the matching number of $G$. In this paper, we show that there is a small constant $c>0$, such that the following decision problem is NP-complete: given a graph $G$ and $k\le \frac{\left|V\right|}{2}$, check whether there is a maximum matching $F$ in $G$, such that $|\nu (G\setminus F)-k|\le c\cdot \left|V\right|$. Note that when $c=1$, this problem is polynomial time solvable as we observe in the paper. Since in any graph $G$, we have $L\left(G\right)\le 2\ell \left(G\right)$, any polynomial time algorithm constructing a maximum matching of a graph is a 2-approximation algorithm for $\ell \left(G\right)$ and $\frac{1}{2}$-approximation algorithm for $L\left(G\right)$. We complement these observations by presenting two inapproximability results for $\ell \left(G\right)$ and $L\left(G\right)$.