Well-order a flame

TL;DR

This paper characterizes infinite flames in rooted digraphs, showing they can be constructed step-by-step starting from an empty graph, extending finite concepts to infinite cases.

Contribution

It provides a transfinite construction method for infinite flames, generalizing finite results and connecting to classical theorems in graph theory.

Findings

Infinite flames can be built transfinitely from empty graphs.

Every intermediate graph in the construction remains a flame.

The approach generalizes finite flame properties to infinite digraphs.

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 Lov\'asz 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. Szeszl\'er later proved a common generalisation of Lov\'asz' 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.