In a graph G = (V, E) a subset F of V is a d-extensible set if for any stable set S of F with |S|= d it exists a stable set S' of V-F F such that SUS' is a maximum stable set of G. We say that S can be extended to a maximum stable set with vertices outside of F, shortly S can be extended. We present a characterization of the d-extensible sets for the class of bipartite graphs and we determine, for some values of d, the maximum cardinality of a d-extensible set. We also determine the maximum cardinality of a d-extensible set for a subclass of arborescences where any vertex x of V belongs to a maximum stable set. This problem has some applications in reliability where we consider vertex failures in the graph G : a d-extensible set is the set of vertices that are not protected.