Reconfiguration problems involve determining whether two given configurations can be transformed into each other under specific rules. The Token Sliding problem asks whether, given two different set of tokens on vertices of a graph $G$, we can transform one into the other by sliding tokens step-by-step along edges of $G$ such that each resulting set of tokens forms an independent set in $G$. Recently, Ito et al. [MFCS 2022] introduced a directed variant of this problem. They showed that for general oriented graphs (i.e., graphs where no pair of vertices can have directed edges in both directions), the problem remains $\mathsf{PSPACE}$-complete, and is solvable in polynomial time on oriented trees. In this paper, we further investigate the Token Sliding problem on various oriented graph classes. We show that the problem remains $\mathsf{PSPACE}$-complete for oriented split graphs, bipartite graphs and bounded treewidth graphs. Additionally, we present polynomial-time algorithms for solving the problem on oriented cycles and cographs.

翻译：重构问题涉及确定两个给定配置是否能在特定规则下相互转换。令牌滑动问题询问：给定图$G$顶点上的两个不同令牌集合，能否通过沿$G$的边逐步滑动令牌，使得每次生成的令牌集合构成$G$中的一个独立集，从而将一个集合转换为另一个集合。最近，Ito等人[MFCS 2022]引入了该问题的一个有向变体。他们证明，对于一般定向图（即任意一对顶点之间不能存在双向有向边的图），该问题仍为$\mathsf{PSPACE}$-完全，并且在定向树上可在多项式时间内求解。在本文中，我们进一步研究了各类定向图上的令牌滑动问题。我们证明，该问题对于定向分裂图、二分图和有界树宽图仍为$\mathsf{PSPACE}$-完全。此外，我们提出了在定向环和余图上求解该问题的多项式时间算法。