Second kind maximum matching graph

The second kind maximum matching graph M 2 ( G ) of a graph G is the graph whose vertices are the maximum matchings of G such that two vertices M 1 and M 2 of M 2 ( G ) are adjacent if and only if the symmetric difference of M 1 and M 2 induces either a cycle or a path of length 2. In this paper, we prove that the class of second kind maximum matching graphs has no forbidden induced subgraphs, and we characterize the graphs whose second kind maximum matching graphs are trees, or cycles, or complete graphs.