A closure concept in factor-critical graphs

A graph G is called n-factor-critical if the removal of every set of n vertices results in a~graph with a~1-factor. We prove the following theorem: Let G be a~graph and let x be a~locally n-connected vertex. Let {u,v} be a~pair of vertices in V(G)−{x} such that uv∉E(G), x∈N_G(u)∩N_G(v), and N_G(x)⊂N_G(u)∪N_G(v)∪{u,v}. Then G is n-factor-critical if and only if G+uv is n-factor-critical.