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Well-order a flame

Zsuzsanna Jankó, Attila Joó

TL;DR

The paper addresses constructing infinite flames in $r$-rooted digraphs, defining flames via edge-disjoint $rv$-paths that cover ingoing edges and preserving local edge-connectivity from the root. It develops a robust toolkit of fillable and $v$-tight sets, the transfinite assembly of witnesses, and an augmenting-path framework to extend path systems. The main contribution is a constructive transfinite theorem: every (possibly infinite) flame can be built by a well-ordered sequence of single-edge additions, with each initial segment remaining a flame; finite cases recover the greedoid structure of flame subgraphs. This work connects to Edmonds’ disjoint-arborescence theorem and provides a canonical, incremental construction of infinite flame subgraphs, with potential implications for understanding global connectivity from a root in infinite networks. The methods blend path-system witnesses, transfinite recursion, and tightness arguments to achieve a comprehensive, constructive characterization of flames.

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.

Well-order a flame

TL;DR

The paper addresses constructing infinite flames in -rooted digraphs, defining flames via edge-disjoint -paths that cover ingoing edges and preserving local edge-connectivity from the root. It develops a robust toolkit of fillable and -tight sets, the transfinite assembly of witnesses, and an augmenting-path framework to extend path systems. The main contribution is a constructive transfinite theorem: every (possibly infinite) flame can be built by a well-ordered sequence of single-edge additions, with each initial segment remaining a flame; finite cases recover the greedoid structure of flame subgraphs. This work connects to Edmonds’ disjoint-arborescence theorem and provides a canonical, incremental construction of infinite flame subgraphs, with potential implications for understanding global connectivity from a root in infinite networks. The methods blend path-system witnesses, transfinite recursion, and tightness arguments to achieve a comprehensive, constructive characterization of flames.

Abstract

An -rooted (possibly infinite) digraph is a flame if for every there exists a set of edge-disjoint paths from to in that covers all ingoing edges of . 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.
Paper Structure (4 sections, 15 theorems, 13 equations, 2 figures)

This paper contains 4 sections, 15 theorems, 13 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation and terminology
  3. Preparations
  4. Proof of Theorem \ref{['thm: main']}

Key Result

Theorem 1.1

Let $D=(V,E)$ be a finite $r$-rooted digraph and $m:=\max_{v\in V\setminus \{ r \}}\lambda_D(r,v)$. Then there are $F_1 \subseteq F_2 \subseteq \cdots \subseteq F_m\subseteq E$ such that for every $i\in [m]$: $D_i:=(V,F_i)$ is a flame with $\lambda_{D_i}(r,v)=\min \{ \lambda_{D}(r,v), i \}$ for $v\i

Figures (2)

  • Figure 1: The construction of the paths $Q_e^{\beta+1}$ for $e\in \delta_D(Y_{\beta+1})$.
  • Figure 2: The tightness of $T$ in Lemma \ref{['lem: char no single']}

Theorems & Definitions (29)

  • Theorem 1.1: szeszler2025some
  • Theorem 1.2
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • proof
  • proof
  • Lemma 3.4
  • proof
  • Corollary 3.5
  • ...and 19 more