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 r-detachment or f-detachment. We solve a problem posed by Nash-Williams in [2] by obtaining analogous results for non-separable detachments of graphs.