An arborescence, which is a directed analogue of a spanning tree in an undirected graph, is one of the most fundamental combinatorial objects in a digraph. In this paper, we study arborescences in digraphs from the viewpoint of combinatorial reconfiguration, which is the field where we study reachability between two configurations of some combinatorial objects via some specified operations. Especially, we consider reconfiguration problems for time-respecting arborescences, which were introduced by Kempe, Kleinberg, and Kumar. We first prove that if the roots of the initial and target time-respecting arborescences are the same, then the target arborescence is always reachable from the initial one and we can find a shortest reconfiguration sequence in polynomial time. Furthermore, we show if the roots are not the same, then the target arborescence may not be reachable from the initial one. On the other hand, we show that we can determine whether the target arborescence is reachable form the initial one in polynomial time. Finally, we prove that it is NP-hard to find a shortest reconfiguration sequence in the case where the roots are not the same. Our results show an interesting contrast to the previous results for (ordinary) arborescences reconfiguration problems.

翻译：树形图作为有向图中无向生成树的模拟，是数字图最基础的组合对象之一。本文从组合重构的视角研究有向图中的树形图——该领域通过特定操作探究组合对象两种配置间的可达性。我们重点关注Kempe、Kleinberg和Kumar提出的时间感知树形图重构问题。首先证明：若初始与目标时间感知树形图的根节点相同，则目标树形图始终可从初始状态到达，且可在多项式时间内找到最短重构序列。进一步研究表明：当根节点不同时，目标树形图可能无法从初始状态到达；但可多项式时间内判定其可达性。最后证明：在根节点不同情形下，寻找最短重构序列是NP难问题。这些结果与（普通）树形图重构问题的现有结论形成鲜明对比。