Galaxy cutsets in graphs

Given a network G=(V,E), we say that a subset of vertices S⊆V has radius r if it is spanned by a tree of depth at most r. We are interested in determining whether G has a cutset that can be written as the union of k sets of radius r. This generalizes the notion of k-vertex connectivity, since in the special case r=0, a set spanned by a tree of depth at most r is a single vertex.