Given a graph $G$ and two independent sets of $G$, the independent set reconfiguration problem asks whether one independent set can be transformed into the other by moving a single vertex at a time, such that at each intermediate step we have an independent set of $G$. We study the complexity of this problem for $H$-free graphs under the token sliding and token jumping rule. Our contribution is twofold. First, we prove a reconfiguration analogue of Alekseev's theorem, showing that the problem is PSPACE-complete unless $H$ is a path or a subdivision of the claw. We then show that under the token sliding rule, the problem admits a polynomial-time algorithm if the input graph is fork-free.

翻译：给定图 $G$ 及其两个独立集，独立集重配置问题探讨是否可以通过每次移动一个顶点，将其中一个独立集逐步转换为另一个独立集，且每一步中间结果均为 $G$ 的独立集。我们研究了在令牌滑动和令牌跳跃规则下，该问题在 $H$-自由图中的复杂性。本文贡献有两方面：首先，我们证明了阿列克谢耶夫定理的重配置版本，表明除非 $H$ 为路径或爪形图的细分图，否则该问题是PSPACE完全的；其次，我们证明在令牌滑动规则下，若输入图不含叉形图，则该问题存在多项式时间算法。