On almost 1-extendable graphs

A graph G is 1-extendable or almost 1-extendable if every edge is contained in a perfect or almost perfect matching of G, respectively. Let d >= 3 be an integer, and let G be a graph of order n with exactly one odd component such that the degree of each vertex is either d or d + 1. If G is not almost 1-extendable, then we prove that n >= 2d + 5. In the special case that d >= 4 is even and G is a d-regular graph, we obtain the better bound n >= 3d + 5. Examples will show that the given bounds are best possible.