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arXiv · 2003.06178

Enlarging vertex-flames in countable digraphs

Joshua Erde·J. Pascal Gollin·Attila Joó
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Abstract

A rooted digraph is a vertex-flame if for every vertex $v$ there is a set of internally disjoint directed paths from the root to $v$ whose set of terminal edges covers all ingoing edges of $v$. It was shown by Lov\'{a}sz that every finite rooted digraph admits a spanning subdigraph which is a vertex-flame and large, where the latter means that it preserves the local connectivity to each vertex from the root. A structural generalisation of vertex-flames and largeness to infinite digraphs was given by the third author and the analogue of Lov\'{a}sz' result for countable digraphs was shown. We strengthen this result by proving that in every countable rooted digraph each vertex-flame can be extended to a large vertex-flame.

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Joshua Erde, J. Pascal Gollin, Attila Joó. 2020-03-13. Enlarging vertex-flames in countable digraphs. https://arxiv.org/abs/2003.06178

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