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 $\lambda(r,v)$ from $r$ to $v$ is equal to $\varrho_F(v)$, the in-degree of $v$. It is a classic, simple and beautiful result of Lov\'asz that every digraph $D$ with a root node $r$ has a spanning subgraph $F$ that is a flame and the $\lambda(r,v)$ 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 Lov\'asz's above mentioned theorem and Edmonds' classic disjoint arborescences theorem.