Reachability in arborescence packings

Fortier et al. \cite{fklst} proposed several research problems on packing arborescences. Some of them were settled in that paper and others were solved later by Matsuoka and Tanigawa \cite{mt} and Gao and Yang \cite{gy}. The last open problem will be settled in this paper. We show how to turn an inductive idea used in the last two articles into a simple proof technique that allows to relate previous results on arborescence packings. We show how a strong version of Edmonds' theorem \cite{e} on packing spanning arborescences implies Kamiyama, Katoh and Takizawa's result \cite{kkt} on packing reachability arborescences and how Durand de Gevigney, Nguyen and Szigeti's theorem \cite{dns} on matroid-based packing of arborescences implies Kir\'aly's result \cite{k} on matroid-reachability-based packing of arborescences. Finally, we deduce a new result on matroid-reachability-based packing of mixed hyperarborescences from a theorem on matroid-based packing of mixed hyperarborescences due to Fortier et al. \cite{fklst}. All the proofs provide efficient algorithms to find a solution to the corresponding problems.