We propose a novel generalization of Independent Set Reconfiguration (ISR): Connected Components Reconfiguration (CCR). In CCR, we are given a graph $G$, two vertex subsets $A$ and $B$, and a multiset $\mathcal{M}$ of positive integers. The question is whether $A$ and $B$ are reconfigurable under a certain rule, while ensuring that each vertex subset induces connected components whose sizes match the multiset $\mathcal{M}$. ISR is a special case of CCR where $\mathcal{M}$ only contains 1. We also propose new reconfiguration rules: component jumping (CJ) and component sliding (CS), which regard connected components as tokens. Since CCR generalizes ISR, the problem is PSPACE-complete. In contrast, we show three positive results: First, CCR-CS and CCR-CJ are solvable in linear and quadratic time, respectively, when $G$ is a path. Second, we show that CCR-CS is solvable in linear time for cographs. Third, when $\mathcal{M}$ contains only the same elements (i.e., all connected components have the same size), we show that CCR-CJ is solvable in linear time if $G$ is chordal. The second and third results generalize known results for ISR and exhibit an interesting difference between the reconfiguration rules.

翻译：我们提出了一种独立集重配置（ISR）的新颖推广：连通分量重配置（CCR）。在CCR中，给定一个图$G$、两个顶点子集$A$和$B$，以及一个正整数多重集$\mathcal{M}$。问题在于$A$和$B$是否能在特定规则下重配置，同时确保每个顶点子集诱导出的连通分量规模与多重集$\mathcal{M}$相匹配。ISR是CCR的一个特例，其中$\mathcal{M}$仅包含1。我们还提出了新的重配置规则：分量跳跃（CJ）和分量滑动（CS），它们将连通分量视为令牌。由于CCR推广了ISR，该问题是PSPACE完全的。与此相对，我们展示了三个积极结果：首先，当$G$是一条路径时，CCR-CS和CCR-CJ分别在线性时间和二次时间内可解。其次，我们证明对于补图，CCR-CS在线性时间内可解。第三，当$\mathcal{M}$仅包含相同元素（即所有连通分量具有相同规模）时，我们证明若$G$是弦图，则CCR-CJ在线性时间内可解。第二和第三个结果推广了ISR的已知结论，并揭示了重配置规则之间一个有趣的差异。