Well-order a flame

Abstract: An $r$-rooted (possibly infinite) digraph $ D=(V,E) $ is a flame if for every $ v\in V\setminus { r } $ there exists a set of edge-disjoint paths from $r$ to $v$ in $D$ that covers all ingoing edges of $ v $. Flames were first studied by Lovász in his investigation of edge-minimal subgraphs of a rooted digraph that preserve all the local edge-connectivities from the root. He showed that these subgraphs are always flames. Szeszlér later proved a common generalisation of Lovász' result and Edmonds' disjoint arborescence theorem. In this paper we focus on infinite flames and prove the following constructive characterisation. Every (possibly infinite) flame can be constructed transfinitely, starting from the empty edge set and adding a single edge at each step in such a way that every intermediate digraph is again a flame.