Non-adjacent vertices

Let G be a connected graph, and assume that every matching in G can be extended to a perfect matching; such a graph is called randomly matchable. Prove that the only randomly matchable graphs on 2n vertices are the graphs Kn,n and K2n; see [Sum79] and [LePP84]. Hint: Show first that G has to be 2-connected. If G is bipartite and contains non-adjacent vertices s and t which are in different parts of G, consider a path (of odd length) from s to t and construct a matching whose only exposed vertices are s and t. Finally, assume that G is not bipartite. Prove that each vertex is contained in a cycle of odd length and that any two vertices are connected by a path of odd length; then proceed as in the bipartite case.