Minimal graphs for matching extensions

Given a positive integer n we find a graph G = (V,E) on |V | = n vertices with a minimum number of edges such that for any pair of non adjacent vertices x, y the graph G − x − y contains a (almost) perfect matching M . Intuitively the non edge xy and M form a (almost) perfect matching of G. Similarly we determine a graph G = (V,E) with a minimum number of edges such that for any matching M̄ of the complement Ḡ of G with size bn2 c − 1, G− V (M̄) contains an edge e. Here M̄ and the edge e of G form a (almost) perfect matching of Ḡ. We characterize these minimal graphs for all values of n.