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：树形结构（arborescence）是有向图中无向生成树的有向类比，是图中最基本的组合对象之一。本文从组合重配置的视角研究有向图中的树形结构——该领域通过特定操作研究某些组合对象的两个配置之间的可达性。我们特别关注由Kempe、Kleinberg和Kumar提出的尊重时间约束的树形结构的重配置问题。首先证明，若初始与目标请求时间的树形结构的根节点相同，则目标树形结构始终可从初始结构通过重配置到达，且可在多项式时间内找到最短重配置序列。进一步证明，若根节点不同，则目标树形结构可能无法从初始结构到达。另一方面，我们表明在多项式时间内可判断目标树形结构是否从初始结构可达。最后证明，当根节点不同时，寻找最短重配置序列是NP-困难的。我们的结果与（普通）树形结构重配置问题的现有结论形成了有趣的对比。