Introduction. A graph G = {N, E} is taken to be a finite set of nodes N together with a set of distinct edges E which are unordered pairs of distinct nodes. A matching M of the graph is a subset of the edges E with the property that no two edges of M are incident at a node. A matching M is perfect if every node is incident to an edge of M. For any finite set X, let IXI denote its cardinality. If S c N let G(N S) be the subgraph of G consisting of nodes N S (nodes of N but not S) and all edges of E which only join nodes of N S. For any subgraph H of G, let o(H) be the number of connected components of H each of which consists of an odd number of nodes. If, for some S c N, o(G(N S)) > ISI, we say that the set S is an imperfectable set of nodes of G.