Matroidal derivation of Edmonds’ arborescence-packing theorem

Determine whether there exists a matroidal theorem or framework whose specialization implies Edmonds’ arborescence-packing theorem, namely, that a digraph D=(V,A) with root-node r0 contains k arc-disjoint spanning r0-arborescences if and only if D is rooted k-arc-connected.

Background

A central theme of the paper is that many fundamental results on packing and covering trees in undirected graphs follow from matroidal theorems (e.g., the tree-packing theorem of Tutte and Nash-Williams has matroidal proofs and generalizations).

In contrast, in the directed setting, although Edmonds’ arborescence-packing theorem is the direct analogue, the authors emphasize that no matroidal result is currently known to imply it. Establishing such a derivation would unify the directed theory with the matroidal framework that successfully explains the undirected case.