Large flames in rooted acyclic digraphs without backward-infinite paths

TL;DR

This paper extends key theorems about flames in finite rooted digraphs to certain infinite acyclic digraphs without backward-infinite paths, revealing their matroid and greedoid structures.

Contribution

It generalizes Lovász's and Szeszlér's theorems to infinite acyclic digraphs lacking backward-infinite paths, establishing their greedoid and matroid properties.

Findings

Flame subgraphs form a greedoid in infinite acyclic digraphs.

Bases of this greedoid are matroids under the no backward-infinite path condition.

Infinite generalizations of Lovász's theorem hold for these digraphs.

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.