Highly edge-connected detachments of graphs and digraphs

Let G = (V,E) be a graph or digraph and r : V → Z+. An r‐detachment of G is a graph H obtained by ‘splitting’ each vertex ν ∈ V into r(ν) vertices. The vertices ν1,…,νr(ν) obtained by splitting ν are called the pieces of ν in H. Every edge uν ∈ E corresponds to an edge of H connecting some piece of u to some piece of ν. Crispin Nash‐Williams 9 gave necessary and sufficient conditions for a graph to have a k‐edge‐connected r‐detachment. He also solved the version where the degrees of all the pieces are specified. In this paper, we solve the same problems for directed graphs. We also give a simple and self‐contained new proof for the undirected result. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 67–77, 2003