On Some Algorithmic and Structural Results on Flames
Dávid Szeszlér
TL;DR
This work addresses finding minimum-weight maximal flames in rooted digraphs, a structure that preserves local rooted edge-connectivity values $λ_D(r,v)$ via the flame condition $λ_F(r,v)=ρ_F(v)$. It shows strong polynomial solvability for acyclic graphs by reducing to a base problem on a sum of gammoid matroids $igoplus_{v∈V} ext{G}_D(v)$ and applying a matroid greedy algorithm, achieving $O(|E|^2)$ time. Additionally, it establishes a decomposition of flames into a chain of edge-disjoint branchings and proves a central theorem that combines Lovász's flame result with Edmonds' disjoint arborescences, providing a unified structural framework and a path toward broader generalizations. The results highlight deep connections between flames, Greedoid and Matroid theory, and classical connectivity theorems, with open questions about broader graph classes and general complexity beyond acyclic graphs.
Abstract
A directed graph $F$ with a root node $r$ is called a flame if for every vertex $v$ other than $r$ the local edge-connectivity value $λ(r,v)$ from $r$ to $v$ is equal to $\varrho_F(v)$, the in-degree of $v$. It is a classic, simple and beautiful result of Lovász that every digraph $D$ with a root node $r$ has a spanning subgraph $F$ that is a flame and the $λ(r,v)$ values are the same in $F$ as in $D$ for every vertex $v$ other than $r$. However, the complexity of finding the minimum weight of such a subgraph is open. In this paper we prove that this problem is solvable in strongly polynomial time for acyclic digraphs. Besides that, we prove a decomposition result of flames into a chain of smaller flames via edge-disjoint branchings and use this to prove a common generalization of Lovász's above mentioned theorem and Edmonds' classic disjoint arborescences theorem.