Minimal graphs for matching extensions

Abstract

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 ⌊n2⌋−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.