Disjoint paths in arborescences

An arborescence in a digraph is a tree directed away from its root. A classical theorem of Edmonds characterizes which digraphs have image arc-disjoint arborescences rooted at r. A similar theorem of Menger guarantees that image strongly arc disjoint image-paths exist for every vertex image, where “strongly” means that no two paths contain a pair of symmetric arcs. We prove that if a directed graph D contains two arc-disjoint spanning arborescences rooted at r, then D contains two such arborences with the property that for every node image the paths from r to image in the two arborences satisfy Mengerʹs theorem.