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arXiv · 2511.22140

Large flames in rooted acyclic digraphs without backward-infinite paths

Attila Joó·Qiuzhenyu Tao
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Abstract

An $r$-rooted digraph is a flame if for each non-root vertex $v$, there is a set of edge-disjoint directed paths from $r$ to $v$ that covers all ingoing edges of $v$. The study of flames was initiated by Lov\'asz, who showed that in a finite rooted digraph, the edge-minimal subgraphs that preserve all local edge-connectivities from the root are always flames. It is known that the edge sets of the flame subgraphs of any finite rooted digraph form a greedoid. Szeszl\'er showed recently that if the digraph is acyclic, then the bases of this greedoid are the bases of a matroid. We show that a suitable formulation of Szeszl\'er's theorem is valid for infinite digraphs under the additional assumption that there are no backward-infinite directed paths (which assumption is indeed essential). We also prove that the ''correct'' infinite generalisation of Lov\'asz's theorem also holds for this class of digraphs.

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Attila Joó, Qiuzhenyu Tao. 2025-11-27. Large flames in rooted acyclic digraphs without backward-infinite paths. https://arxiv.org/abs/2511.22140

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