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 matching discovered independently by J. P. Uhry and E. Balas. This article is motivated by a property of fractional matchings discovered by J.P. Uhry[8] and E. Balas[2]. We characterize the class of graphs to which the property can be naturally extended. We consider finite graphs without loops or multiple edges. Let K2 denote the complete graph of order two and Cn denote the cycle of order n. Let {Ai|i ∈ I} be a set of graphs. A subgraph F of a graph G is called an {Aı|i ∈ I}-subgraph if each component of F is isomorphic to one of {Ai|i ∈ I}. A component of F that is isomorphic to Ai is called an Ai-component. A spanning {Ai|i ∈ I}-subgraph is called an {Ai|i ∈ I}-factor. An {Ai|i ∈ I}subgraph F is said to be maximum if there is no {Ai|i ∈ I}-subgraph F ′ such that |V (F ′)| > |V (F )|. A fractional matching is a {K2, CC2n+1|n ≥ 1}subgraph, and J.P. Uhry and E. Balas proved the following theorem: Theorem 1 ([8,2]). Let F be a maximum fractional matching of a graph G having minimum number of odd cycles. Then a maximum matching of F if also a maximum matching of G.