The independent set reconfiguration problem (ISReconf) is the problem of determining, for given independent sets I_s and I_t of a graph G, whether I_s can be transformed into I_t by repeatedly applying a prescribed reconfiguration rule that transforms an independent set to another. As reconfiguration rules for the ISReconf, the Token Sliding (TS) model and the Token Jumping (TJ) model are commonly considered. While the TJ model admits the addition of any vertex (as far as the addition yields an independent set), the TS model admits the addition of only a neighbor of the removed vertex. It is known that the complexity status of the ISReconf differs between the TS and TJ models for some graph classes. In this paper, we analyze how changes in reconfiguration rules affect the computational complexity of reconfiguration problems. To this end, we generalize the TS and TJ models to a unified reconfiguration rule, called the k-Jump model, which admits the addition of a vertex within distance k from the removed vertex. Then, the TS and TJ models are the 1-Jump and D(G)-Jump models, respectively, where D(G) denotes the diameter of a connected graph G. We give the following three results: First, we show that the computational complexity of the ISReconf under the k-Jump model for general graphs is equivalent for all k >= 3. Second, we present a polynomial-time algorithm to solve the ISReconf under the 2-Jump model for split graphs. We note that the ISReconf under the 1-Jump (i.e., TS) model is PSPACE-complete for split graphs, and hence the complexity status of the ISReconf differs between k = 1 and k = 2. Third, we consider the optimization variant of the ISReconf, which computes the minimum number of steps of any transformation between Is and It. We prove that this optimization variant under the k-Jump model is NP-complete for chordal graphs of diameter at most 2k + 1, for any k >=3.

翻译：独立集重构问题（ISReconf）是判定对于给定图G的独立集I_s和I_t，能否通过反复应用规定的重构规则将I_s转换为I_t的问题。针对ISReconf的重构规则，通常考虑令牌滑动（TS）模型和令牌跳跃（TJ）模型。TJ模型允许添加任意顶点（只要添加后仍为独立集），而TS模型仅允许添加被移除顶点的邻居顶点。已知对于某些图类，TS与TJ模型下的ISReconf复杂度状态存在差异。本文分析重构规则变化如何影响重构问题的计算复杂度。为此，我们将TS和TJ模型推广为统一的重构规则——k-跳模型，该模型允许添加与被移除顶点距离不超过k的顶点。此时TS模型和TJ模型分别对应1-跳模型和D(G)-跳模型，其中D(G)表示连通图G的直径。我们给出以下三个结果：首先，证明对于一般图，k-跳模型下ISReconf的计算复杂度对所有k≥3等价。其次，针对分裂图提出解决2-跳模型下ISReconf的多项式时间算法。注意到1-跳（即TS）模型下分裂图的ISReconf复杂度为PSPACE完全，因此k=1与k=2的复杂度状态存在差异。第三，考虑计算I_s与I_t之间任意转换所需最少步数的ISReconf优化变体。证明对于任意k≥3，直径不超过2k+1的弦图上，k-跳模型下的该优化变体为NP完全。