Matroid-rooted packing of arborescences

The problem of matroid-based packing of arborescences was introduced and solved in Durand de Gevigney et al. (SIAM J Discret Math 27(1):567-574) . Frank (In personal communication) reformulated the problem in an extended framework. We proved in Fortier et al. (J Graph Theory 93(2):230-252) that the problem of matroid-based packing of spanning arborescences is NP-complete in the extended framework. Here we show a characterization of the existence of a matroid-based packing of spanning arborescences in the original framework. This leads us to the introduction of a new problem on packing of arborescences with a new matroid constraint. We characterize mixed graphs having a matroid-rooted, k-regular, (f, g)-bounded packing of mixed arborescences, that is, a packing of mixed arborescences such that their roots form a basis in a given matroid, each vertex belongs to exactly k of them and each vertex v is the root of least f(v) and at most g(v) of them. We also characterize dypergraphs having a matroid-rooted, k-regular, (f, g)-bounded packing of hyperarborescences.