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Highly edge-connected detachments of graphs and digraphs. (English) Zbl 1014.05043
Let \(G=(V,E)\) be a graph or digraph and \(r:V\to Z_+\). An \(r\)-detachment of \(G\) is a graph \(H\) obtained by ‘splitting’ each vertex \(v\in V\) into \(r(v)\) vertices. The vertices \(v_1,v_2,\dots,v_{r(v)}\) obtained by splitting \(v\) are called the pieces of \(v\) in \(H\). Every edge \(uv\in E\) corresponds to an edge of \(H\) connecting some piece of \(u\) to some piece of \(v\). C. St. J. A. Nash-Williams [J. Lond. Math. Soc., II. Ser. 31, 17-29 (1985; Zbl 0574.05042)] gave necesary and sufficient conditions for a graph to have a \(k\)-edge-connected \(r\)-detachment, and also solved the version where the degrees of all the pieces are specified. In this paper, the authors solve the same problems for directed graphs, and give a simple and self-contained new proof for the undirected result.

05C40 Connectivity
05C20 Directed graphs (digraphs), tournaments
Full Text: DOI
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