Is the number of exposed vertices equal for different maximum matchings?

Question

Given two maximum matchings $M$ and $M'$ on a given undirected graph $G=(V,E)$ (not necessarily bipartite), with $M$ and $M'$ having, respectively, $p$ and $p'$ exposed vertices (i.e. vertices not used by $M$ or $M'$ accordingly), must we have $p=p'$?

Thanks!

Answer 1

Suppose the matchings $M$ and $M'$ cover $e$ edges then they will cover $2e$ vertices, so \begin{eqnarray*} 2e+p=2e+p'. \end{eqnarray*}