Greedoids from flames

A digraph $D$ with $r\in V(D)$ is an $r$-flame if for every ${v\in V(D)-r}$, the in-degree of $v$ is equal to the local edge-connectivity $\lambda_D(r,v)$. We show that for every digraph $D$ and $r\in V(D)$, the edge sets of the $r$-flame subgraphs of $D$ form a greedoid. Our method yields a new proof of Lov\'asz' theorem stating: for every digraph $D$ and $r\in V(D)$, there is an $r$-flame subdigraph $F$ of $D$ such that $\lambda_F(r,v) =\lambda_D(r,v)$ for $v\in V(D)-r$. We also give a strongly polynomial algorithm to find such an $F$ working with a fractional generalization of Lov\'asz' theorem.