An arborescence in a digraph is an acyclic arc subset in which every vertex execpt a root has exactly one incoming arc. In this paper, we reveal the reconfigurability of the union of $k$ arborescences for fixed $k$ in the following sense: for any pair of arc subsets that can be partitioned into $k$ arborescences, one can be transformed into the other by exchanging arcs one by one so that every intermediate arc subset can also be partitioned into $k$ arborescences. This generalizes the result by Ito et al. (2023), who showed the case with $k=1$. Since the union of $k$ arborescences can be represented as a common matroid basis of two matroids, our result gives a new non-trivial example of matroid pairs for which two common bases are always reconfigurable to each other.

翻译：在有向图中，有向树是一个无环的弧子集，其中除根节点外每个顶点恰好有一条入弧。本文揭示了固定整数$k$下$k$棵有向树并集的可重构性，具体表现为：对于任意两个可被划分为$k$棵有向树的弧子集，可以通过逐一交换弧的方式将一个子集转化为另一个，且中间过程中的每个弧子集仍可被划分为$k$棵有向树。该结果推广了Ito等人(2023)关于$k=1$情形的结论。由于$k$棵有向树的并集可表示为两个拟阵的公共基，本文为两个公共基始终可相互重构的拟阵对提供了新的非平凡实例。