Non-separable detachments of graphs (Q1403909)

Let \(G=(V,E)\) be a graph (loops and multiple edges permitted) and let \(r:V\to Z_+\). An \(r\)-detachment of \(G\) is a graph \(H\) whose vertices are obtained by splitting each vertex \(v\in V\) into \(r(v)\) vertices, called the pieces of \(v\); each edge \(uv\in E\) corresponds to an edge of \(H\) joining a piece of \(u\) to a piece of \(v\). An \(r\)-degree specification for \(G\) is a function \(f\) on \(V\) such that \(f(v)\) is a partition of the valence of \(v\) into \(r(v)\) positive integers. An \(f\)-detachment of \(G\) is an \(r\)-detachment in which the valences of the pieces of each \(v\in V\) are given by \(f(v)\). The authors answer a question posed by \textit{C. St. J. A. Nash-Williams} [Surveys in combinatorics 1985, Proc. 10th British Combin. Conf., Glasgow/Scotl. 1985, Lond. Math. Soc. Lect. Note Ser. 103, 137-151 (1985; Zbl 0573.05054)] by giving necessary and sufficient conditions on a graph \(G=(V,E)\) and a function \(r:V\to Z_+\) for the existence of a cut-vertex-free \(r\)-detachment of \(G\). Furthermore, an analogous result is obtained for the existence of a cut-vertex-free \(f\)-detachment.