For a fixed integer $r \geq 1$, a distance-$r$ dominating set of a graph $G = (V, E)$ is a vertex subset $D \subseteq V$ such that every vertex in $V$ is within distance $r$ from some member of $D$. Given two distance-$r$ dominating sets $D_s, D_t$ of $G$, the Distance-$r$ Dominating Set Reconfiguration (D$r$DSR) problem asks if there is a sequence of distance-$r$ dominating sets that transforms $D_s$ into $D_t$ (or vice versa) such that each intermediate member is obtained from its predecessor by applying a given reconfiguration rule exactly once. The problem for $r = 1$ has been well-studied in the literature. We consider D$r$DSR for $r \geq 2$ under two well-known reconfiguration rules: Token Jumping ($\mathsf{TJ}$, which involves replacing a member of the current D$r$DS by a non-member) and Token Sliding ($\mathsf{TS}$, which involves replacing a member of the current D$r$DS by an adjacent non-member). We show that D$r$DSR ($r \geq 2$) is $\mathtt{PSPACE}$-complete under both $\mathsf{TJ}$ and $\mathsf{TS}$ on bipartite graphs, planar graphs of maximum degree six and bounded bandwidth, and chordal graphs. On the positive side, we show that D$r$DSR ($r \geq 2$) can be solved in polynomial time on split graphs and cographs under both $\mathsf{TS}$ and $\mathsf{TJ}$ and on trees and interval graphs under $\mathsf{TJ}$. Along the way, we observe some properties of a shortest reconfiguration sequence in split graphs when $r = 2$, which may be of independent interest.

翻译：对于固定整数 $r \geq 1$，图 $G = (V, E)$ 的一个距离-$r$ 控制集是顶点子集 $D \subseteq V$，使得 $V$ 中的每个顶点与 $D$ 中某成员的距离均不超过 $r$。给定 $G$ 的两个距离-$r$ 控制集 $D_s, D_t$，距离-$r$ 控制集重构（D$r$DSR）问题询问是否存在一系列距离-$r$ 控制集将 $D_s$ 转换为 $D_t$（反之亦然），使得每个中间成员通过恰好一次给定的重构规则从其前驱得到。$r = 1$ 的情形已在文献中得到充分研究。我们考虑在两种经典重构规则下 $r \geq 2$ 的 D$r$DSR 问题：令牌跳跃（$\mathsf{TJ}$，将当前 D$r$DS 的一个成员替换为一个非成员）和令牌滑动（$\mathsf{TS}$，将当前 D$r$DS 的一个成员替换为一个相邻的非成员）。我们证明，在 $\mathsf{TJ}$ 和 $\mathsf{TS}$ 规则下，D$r$DSR（$r \geq 2$）在二部图、最大度为六且带宽有界的平面图以及弦图上均为 $\mathtt{PSPACE}$-完全。在正面结果方面，我们证明 D$r$DSR（$r \geq 2$）在 $\mathsf{TS}$ 和 $\mathsf{TJ}$ 规则下可在多项式时间内解决于分裂图和余图，并在 $\mathsf{TJ}$ 规则下可多项式时间解决于树和区间图。在此过程中，我们观察到 $r = 2$ 时分裂图中最短重构序列的一些性质，这些性质可能具有独立的研究价值。