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Large flames in rooted acyclic digraphs without backward-infinite paths

Attila Joó, Qiuzhenyu Tao

TL;DR

This work extends the theory of flames from finite rooted digraphs to a broad class of infinite acyclic digraphs by identifying backward-infinite paths as the sole obstacle to extending Szeszlér's matroid-based results. It develops a transfinite, well-ordered framework using $v$-linked sets, fillability, and infinite Menger-type theorems to show every flame in such digraphs can be extended to a large flame, and that maximal elements of the associated family $\mathcal{G}(D)$ are large. It also connects these constructions to Lovász's original theorem by establishing a robust infinite generalization of the large-flame property for acyclic graphs without backward-infinite paths. The paper closes with open problems and conjectures aiming to remove the restriction and to further generalize Szeszlér's result to broader infinite settings.

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 Lovász, 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. Szeszlér 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 Szeszlér'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 Lovász's theorem also holds for this class of digraphs.

Large flames in rooted acyclic digraphs without backward-infinite paths

TL;DR

This work extends the theory of flames from finite rooted digraphs to a broad class of infinite acyclic digraphs by identifying backward-infinite paths as the sole obstacle to extending Szeszlér's matroid-based results. It develops a transfinite, well-ordered framework using -linked sets, fillability, and infinite Menger-type theorems to show every flame in such digraphs can be extended to a large flame, and that maximal elements of the associated family are large. It also connects these constructions to Lovász's original theorem by establishing a robust infinite generalization of the large-flame property for acyclic graphs without backward-infinite paths. The paper closes with open problems and conjectures aiming to remove the restriction and to further generalize Szeszlér's result to broader infinite settings.

Abstract

An -rooted digraph is a flame if for each non-root vertex , there is a set of edge-disjoint directed paths from to that covers all ingoing edges of . The study of flames was initiated by Lovász, 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. Szeszlér 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 Szeszlér'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 Lovász's theorem also holds for this class of digraphs.

Paper Structure

This paper contains 3 sections, 15 theorems, 1 equation, 1 figure.

Table of Contents

  1. Introduction
  2. Proof of the main results
  3. Open problems

Key Result

Theorem 1.1

In any finite $r$-rooted digraph $D$, the flames in $D$ form a greedoid.In other words, if $F$ and $F'$ are flames in $D$ with $\left|F \right|<\left|F' \right|$, then there is an $e\in F' \setminus F$ such that $F\cup \{ e \}$ is a flame.

Figures (1)

  • Figure 1: The horizontal edges together with the edge from $r$ to $w$ form a maximal element $B$ of $\mathcal{G}(D)$. Clearly, $B$ is not large because $\lambda_D(r,v_n)=1$ but $\lambda_{D(B)}(r,v_n)=0$ for each $n\in \mathbb{N}$.

Theorems & Definitions (32)

  • Theorem 1.1: Joó, jooGreedoidFlame2021
  • Theorem 1.2: Lovász, lovasz1973connectivity
  • Theorem 1.3: Szeszlér, szeszler2025some
  • Theorem 1.3
  • Theorem 1.3
  • Theorem 2.1: Aharoni and Berger, aharoni2009menger
  • Theorem 2.2: Diestel and Thomassen, diestel2006cantor
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 22 more