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 relation of the cardinalities of a maximum matching and the largest matching in those pairs. It is known that this relation does not exceed 5/4 for any graph. We characterize the class of graphs for which this relation is precisely 5/4. Our characterization implies that these graphs contain a spanning subgraph, every component of which is the minimal graph of this class.