An upper bound for a parameter related to special maximum matching constructing

For a graph G let L(G) and l(G) denote the size of the largest and smallest maximum matching of graphs obtained from G by removing its maximum matchings. We show that L(G) ≤ 2l(G), and L(G) ≤ 3 2 l(G) provided that G contains a perfect matching. We also characterize the class of graphs for with L(G) = 2l(G). Our characterization implies the existence of a polynomial algorithm for testing the property L(G) = 2l(G). Finally we show that it is NP -complete to test whether a graph G containing a perfect matching satisfies L(G) = 3 2 l(G). In the paper graphs are assumed to be finite, undirected, without loops or multiple edges. Let V (G) and E(G) denote the sets of vertices and edges of a graph G, respectively. The length of a path is the number of edges lying on it. A k-path is a path of length k. For a graph G, let V1(G) (or shortly, V1) denote the set of vertices having degree one, and let ν(G) denote the cardinality of the largest matching of G. Define: L(G) ≡ max{ν(G\F )/F is a maximum matching of G}, l(G) ≡ min{ν(G\F )/F is a maximum matching of G}. It is known that L(G) and l(G) are NP -hard calculable even for connected bipartite graphs G with maximum degree three [4], though there are polynomial algorithms which construct a maximum matching F of a tree G such that ν(G\F ) = L(G) or ν(G\F ) = l(G) (to be presented in [5]). The paper is based on the first author’s BS theses completed under supervision of the second author The author is supported by a grant of Armenian National Science and Education Fund