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Let Dgoodbreakinfix=(V+s,A) be a digraph with a designated root vertex s. Edmonds’ seminal result (see J. Edmonds [4]) implies that D has a packing of k spanning s‐arborescences if and only if D has a packing of k(s,t)‐paths for all tgoodbreakinfix∈V, where a packing means arc‐disjoint subgraphs. Let bold-scriptM be a matroid on the set of arcs leaving s. A packing of (s,t)‐paths is called bold-scriptM‐based if their arcs leaving s form a base of bold-scriptM while a packing of s‐arborescences is called bold-scriptM‐based if, for all tgoodbreakinfix∈V, the packing of (s,t)‐paths provided by the arborescences is bold-scriptM‐based. Durand de Gevigney, Nguyen, and Szigeti proved in [3] that D has an bold-scriptM‐based packing of s‐arborescences if and only if D has an bold-scriptM‐based packing of (s,t)‐paths for all tgoodbreakinfix∈V. Bérczi and Frank conjectured that this statement can be strengthened in the sense of Edmonds’ theorem such that each s‐arborescence is required to be spanning. Specifically, they conjectured that D has an bold-scriptM‐based packing of spanning s‐arborescences if and only if D has an bold-scriptM‐based packing of (s,t)‐paths for all tgoodbreakinfix∈V. In this paper we disprove this conjecture in its general form and we prove that the corresponding decision problem is NP‐complete. We also prove that the conjecture holds for several fundamental classes of matroids, such as graphic matroids and transversal matroids. For all the results presented in this paper, the undirected counterpart also holds.

Let us be given a rooted digraph D = (V +s, A) with a designated root vertex s. Edmonds' seminal result [3] states that D has a packing of k spanning s-arborescences if and only if D has a packing of k (s, t)-paths for all t ∈ V , where a packing means arc-disjoint subgraphs.Let M be a matroid on the set of arcs leaving s. A packing of (s, t)-paths is called M-based if their arcs leaving s form a base of M while a packing of sarborescences is called M-based if, for all t ∈ V, the packing of (s, t)-paths provided by the arborescences is M-based. Durand de Gevigney, Nguyen and Szigeti proved in [2] that D has an M-based packing of s-arborescences if and only if D has an M-based packing of (s, t)-paths for all t ∈ V. Bérczi and Frank conjectured that this statement can be strengthened in the sense of Edmonds' theorem such that each s-arborescence is required to be spanning. Specifically, they conjectured that D has an M-based packing of spanning s-arborescences if and only if D has an M-based packing of (s, t)-paths for all t ∈ V .We disprove this conjecture in its general form and we prove that the corresponding decision problem is NP-complete. However, we prove that the conjecture