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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.

##### MSC:
 05C40 Connectivity 05C20 Directed graphs (digraphs), tournaments
##### Keywords:
detachments; edge-connectivity; digraphs
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##### References:
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