In reconfiguration problems, we are given two feasible solutions to a graph problem and asked whether one can be transformed into the other via a sequence of feasible intermediate solutions under a given reconfiguration rule. While earlier work focused on modifying a single element at a time, recent studies have started examining how different rules impact computational complexity. Motivated by recent progress, we study Independent Set Reconfiguration (ISR) and Vertex Cover Reconfiguration (VCR) under the $k$-Token Jumping ($k$-TJ) and $k$-Token Sliding ($k$-TS) models. In $k$-TJ, up to $k$ vertices may be replaced, while $k$-TS additionally requires a perfect matching between removed and added vertices. It is known that the complexity of ISR crucially depends on $k$, ranging from PSPACE-complete and NP-complete to polynomial-time solvable. In this paper, we further explore the gradient of computational complexity of the problems. We first show that ISR under $k$-TJ with $k = |I| - \mu$ remains NP-hard when $\mu$ is any fixed positive integer and the input graph is restricted to graphs of maximum degree 3 or planar graphs of maximum degree 4, where $|I|$ is the size of feasible solutions. In addition, we prove that the problem belongs to NP not only for $\mu=O(1)$ but also for $\mu = O(\log |I|)$. In contrast, we show that VCR under $k$-TJ is in XP when parameterized by $\mu = |S| - k$, where $|S|$ is the size of feasible solutions. Furthermore, we establish the PSPACE-completeness of ISR and VCR under both $k$-TJ and $k$-TS on several graph classes, for fixed $k$ as well as superconstant $k$ relative to the size of feasible solutions.

翻译：在重构问题中，给定图问题的两个可行解，需要判断在特定重构规则下能否通过一系列可行中间解将一个解转换为另一个。早期研究主要关注每次修改单个元素，而近期研究开始探讨不同规则对计算复杂性的影响。受最新进展启发，我们研究了$k$-令牌跳跃（$k$-TJ）和$k$-令牌滑动（$k$-TS）模型下的独立集重构（ISR）与顶点覆盖重构（VCR）问题。在$k$-TJ中，最多可替换$k$个顶点；而$k$-TS额外要求被移除顶点与新增顶点间存在完美匹配。已知ISR的复杂度关键取决于$k$，其范围从PSPACE完全、NP完全到多项式时间可解。本文进一步探究了这些问题计算复杂性的梯度变化。首先证明当$\mu$为任意固定正整数且输入图限制为最大度3的图或最大度4的平面图时（其中$|I|$为可行解规模），$k = |I| - \mu$条件下的$k$-TJ-ISR问题仍为NP困难问题。此外，我们证实该问题不仅对$\mu=O(1)$属于NP类，对$\mu = O(\log |I|)$同样成立。与之相对，我们证明以$\mu = |S| - k$为参数时（$|S|$为可行解规模），$k$-TJ-VCR问题属于XP类。进一步地，我们在多个图类上建立了$k$-TJ与$k$-TS模型下ISR和VCR问题的PSPACE完全性结论，其中$k$既包含固定值也包含相对于可行解规模的超常数情形。