For a fixed positive integer $d \geq 2$, a distance-$d$ independent set (D$d$IS) of a graph is a vertex subset whose distance between any two members is at least $d$. Imagine that there is a token placed on each member of a D$d$IS. Two D$d$ISs are adjacent under Token Sliding ($\mathsf{TS}$) if one can be obtained from the other by moving a token from one vertex to one of its unoccupied adjacent vertices. Under Token Jumping ($\mathsf{TJ}$), the target vertex needs not to be adjacent to the original one. The Distance-$d$ Independent Set Reconfiguration (D$d$ISR) problem under $\mathsf{TS}/\mathsf{TJ}$ asks if there is a corresponding sequence of adjacent D$d$ISs that transforms one given D$d$IS into another. The problem for $d = 2$, also known as the Independent Set Reconfiguration problem, has been well-studied in the literature and its computational complexity on several graph classes has been known. In this paper, we study the computational complexity of D$d$ISR on different graphs under $\mathsf{TS}$ and $\mathsf{TJ}$ for any fixed $d \geq 3$. On chordal graphs, we show that D$d$ISR under $\mathsf{TJ}$ is in $\mathtt{P}$ when $d$ is even and $\mathtt{PSPACE}$-complete when $d$ is odd. On split graphs, there is an interesting complexity dichotomy: D$d$ISR is $\mathtt{PSPACE}$-complete for $d = 2$ but in $\mathtt{P}$ for $d=3$ under $\mathsf{TS}$, while under $\mathsf{TJ}$ it is in $\mathtt{P}$ for $d = 2$ but $\mathtt{PSPACE}$-complete for $d = 3$. Additionally, certain well-known hardness results for $d = 2$ on perfect graphs and planar graphs of maximum degree three and bounded bandwidth can be extended for $d \geq 3$.

翻译：对于固定正整数$d \geq 2$，图的距离$d$独立集（D$d$IS）是指任意两个成员之间距离至少为$d$的顶点子集。假设D$d$IS的每个成员上都放置一个标记。在Token Sliding（$\mathsf{TS}$）规则下，两个D$d$IS相邻当且仅当其中一个可通过将某个标记从其所在顶点移动到一个未被占据的相邻顶点得到。在Token Jumping（$\mathsf{TJ}$）规则下，目标顶点无需与原顶点相邻。距离$d$独立集重配置（D$d$ISR）问题在$\mathsf{TS}/\mathsf{TJ}$规则下询问是否存在一列相邻的D$d$IS序列，将给定的一个D$d$IS变换为另一个。当$d = 2$时，该问题即独立集重配置问题，已在文献中得到充分研究，且其在若干图类上的计算复杂度已被知晓。本文研究对于任意固定$d \geq 3$，在不同图上D$d$ISR在$\mathsf{TS}$和$\mathsf{TJ}$规则下的计算复杂度。在弦图上，我们证明当$d$为偶数时D$d$ISR在$\mathsf{TJ}$规则下属于$\mathtt{P}$，而当$d$为奇数时则为$\mathtt{PSPACE}$-完全问题。在分裂图中存在有趣的复杂度二分性：在$\mathsf{TS}$规则下，D$d$ISR在$d = 2$时为$\mathtt{PSPACE}$-完全，但在$d = 3$时属于$\mathtt{P}$；而在$\mathsf{TJ}$规则下，它在$d = 2$时属于$\mathtt{P}$，却在$d = 3$时为$\mathtt{PSPACE}$-完全。此外，原有的某些关于完美图、最大度为三的平面图以及有界带宽图在$d = 2$时的困难性结果可推广至$d \geq 3$的情形。