Characterization of a class of graphs related to pairs of disjoint matchings

For a given graph consider a pair of disjoint matchings the union of which contains as many edges as possible. Furthermore, consider the ratio of the cardinalities of a maximum matching and the largest matching in those pairs. It is known that for any graph 5 4 is the tight upper bound for this ratio. We characterize the class of graphs for which it is precisely 5 4 . Our characterization implies that these graphs contain a spanning subgraph, every connected component of which is the minimal graph of this class.