Non-separable detachments of graphs

Let G = (V,E) be a graph and r : V → Z+. An r-detachment of G is a graph H obtained by ‘splitting’ each vertex v ∈ V into r(v) vertices, called the pieces of v in H. Every edge uv ∈ E corresponds to an edge of H connecting some piece of u to some piece of v. An r-degree specification for G is a function f on V , such that, for each vertex v ∈ V , f(v) is a partition of d(v) into r(v) positive integers. An f -detachment of G is an r-detachment H in which the degrees in H of the pieces of each v ∈ V are given by f(v). Crispin Nash-Williams [3] obtained necessary and sufficient conditions for a graph to have a k-edge-connected rdetachment or f -detachment. We solve a problem posed by Nash-Williams in [2] by obtaining analogous results for non-separable detachments of graphs.