On Some Algorithmic and Structural Results on Flames

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 ${\lambda }_F\left(r,v\right)$ from $r$ to $v$ is equal to ${\varrho }_F\left(v\right)$, the in-degree of $v$. It is a classic, simple and beautiful result of Lovász [4] that every digraph $D$ with a root node $r$ has a spanning subgraph $F$ that is a flame and the $\lambda \left(r,v\right)$ 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 [3]. 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 edge-disjoint branchings via a chain of smaller flames and use this to prove a common generalization of Lovász's above mentioned theorem and Edmonds' classic disjoint arborescences theorem.