Graphs with the balas—uhry property

We characterize graphs H with the following property: Let G be a graph and F be a subgraph of G such that (i) each component of F is isomorphic to H or K₂, (ii) the order of F is maximum, and (iii) the number of H-components in F is minimum subject to (ii). Then a maximum matching of F is also a maximum matching of G. This result is motivated by an analogous property of fractional matchings discovered independently by J. P. Uhry and E. Balas.