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 ) ≤ 2 l ( G ) , and L ( G ) ≤ 3 2 l ( G ) provided that G contains a perfect matching. We also characterize the class of graphs for which L ( G ) = 2 l ( G ) . Our characterization implies the existence of a polynomial algorithm for testing the property L ( G ) = 2 l ( G ) . Finally we show that it is N P -complete to test whether a graph G containing a perfect matching satisfies L ( G ) = 3 2 l ( G ) .