Packing branchings under cardinality constraints on their root sets

Edmonds' fundamental theorem on arborescences characterizes the existence of $k$ pairwise arc-disjoint spanning with prescribed root sets in a digraph. In this paper, we study problem packing branchings digraphs under cardinality constraints their by arborescence augmentation. Let $D=(V+x,A)$ be digraph, $\mathcal{P}=$ $\{I_{1}, \ldots, I_{l} \}$ partition $[k]$, $c_{1}, c_{l}, c'_{1}, c'_{l}$ nonnegative integers such that $c_{\alpha} \leq c'_{\alpha}$ for $\alpha \in [l]$, $F_{1}, F_{k}$ $x$-arborescences $D$ $\sum_{i I_{\alpha}}d_{F_{i}}^{+}(x)$ $\leq [l]$. We give characterization when can completed to $F^{*}_{1}, F^{*}_{k}$ any $ c_{\alpha} \sum_{i I_{\alpha}}d^{+}_{F^{*}_{i}}(x)$ c'_{\alpha}$.