Packing branchings under cardinality constraints on their root sets

Edmonds’ fundamental theorem on arborescences characterizes the existence of $k$ pairwise arc-disjoint spanning arborescences with prescribed root sets in a digraph. In this paper, we study the problem of packing branchings in digraphs under cardinality constraints on their root sets by arborescence augmentation. Let $D=(V+x,A)$ be a digraph, $\mathcal{P}=$ $\{I_1,\dots ,I_l\}$ be a partition of $[k]$, $c_1,\dots ,c_l,c_1^{\mathrm{\prime }},\dots ,c_l^{\mathrm{\prime }}$ be nonnegative integers such that $c_{\alpha }\le c_{\alpha }^{\mathrm{\prime }}$ for $\alpha \in [l]$,$F_{1}, \ldots, F_{k}$be$k$arc-disjoint$x$-arborescencesin$D$ such that ${\sum }_{i\in I_{\alpha }}{d}_{F_i}^{+}(x)$ $\le c_{\alpha }^{\mathrm{\prime }}$ for $\alpha \in [l]$\mathrm{.}Wegiveacharacterizationonwhen$F_{1}, \ldots, F_{k}$ can be completed to arc-disjoint spanning $x$-arborescences $F^{}_{1}, \ldots, F^{}{k}$suchthatforany$\alpha \in [l]$,$ c{\alpha} \leq \sum_{i \in I_{\alpha}}d^{+}{F^{*}{i}}(x)$ $\le c_{\alpha }^{\mathrm{\prime }}$.