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 frac(5, 4) is the tight upper bound for this ratio. We characterize the class of graphs for which it is precisely frac(5, 4). Our characterization implies that these graphs contain a spanning subgraph, every connected component of which is the minimal graph of this class. © 2008 Elsevier B.V. All rights reserved.
Tserunyan, A. (2009). Characterization of a class of graphs related to pairs of disjoint matchings. Discrete Mathematics, 309(4), 693–713. https://doi.org/10.1016/j.disc.2008.01.004