On upper bounds for parameters related to the construction of special maximum matchings

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) ≤ 2l(G), and L(G) ≤ 3 2 l(G) provided that G contains a perfect matching. We also characterize the class of graphs for with 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).