For a fixed integer $r \geq 1$, a distance-$r$ dominating set (D$r$DS) 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 D$r$DSs $D_s, D_t$ of $G$, the Distance-$r$ Dominating Set Reconfiguration (D$r$DSR) problem asks if there is a sequence of D$r$DSs 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). It is known that under any of $\mathsf{TS}$ and $\mathsf{TJ}$, the problem on split graphs is $\mathtt{PSPACE}$-complete for $r = 1$. We show that for $r \geq 2$, the problem is in $\mathtt{P}$, resulting in an interesting complexity dichotomy. Along the way, we prove some non-trivial bounds on the length of a shortest reconfiguration sequence on split graphs when $r = 2$ which may be of independent interest. Additionally, we design a linear-time algorithm under $\mathsf{TJ}$ on trees. On the negative side, we show that D$r$DSR for $r \geq 1$ on planar graphs of maximum degree three and bounded bandwidth is $\mathtt{PSPACE}$-complete, improving the degree bound of previously known results. We also show that the known $\mathtt{PSPACE}$-completeness results under $\mathsf{TS}$ and $\mathsf{TJ}$ for $r = 1$ on bipartite graphs and chordal graphs can be extended for $r \geq 2$.

翻译：对固定整数 $r \geq 1$，图 $G = (V, E)$ 的距离-$r$ 控制集 (D$r$DS) 是顶点子集 $D \subseteq V$，使得 $V$ 中每个顶点与 $D$ 中某成员的距离不超过 $r$。给定 $G$ 的两个 D$r$DS $D_s, D_t$，距离-$r$ 控制集重配置问题 (D$r$DSR) 询问是否存在一个 D$r$DS 序列将 $D_s$ 变换为 $D_t$（或反之），使得每个中间成员通过应用给定重配置规则恰好一次由其前驱得到。$r=1$ 的情形已在文献中得到充分研究。我们考虑 $r \geq 2$ 时在两种常见重配置规则下的 D$r$DSR：令牌跳跃 ($\mathsf{TJ}$，涉及将当前 D$r$DS 的某成员替换为一个非成员) 和令牌滑动 ($\mathsf{TS}$，涉及将当前 D$r$DS 的某成员替换为一个相邻非成员)。已知在 $\mathsf{TS}$ 或 $\mathsf{TJ}$ 规则下，$r=1$ 时分裂图上的问题是 $\mathtt{PSPACE}$-完备的。我们证明 $r \geq 2$ 时该问题属于 $\mathtt{P}$，由此得到一个有趣的复杂度二分性。在此过程中，我们证明了 $r=2$ 时分裂图上最短重配置序列长度的若干非平凡界，这可能具有独立意义。此外，我们在树上设计了 $\mathsf{TJ}$ 规则下的线性时间算法。从负面结果看，我们证明最大度为三且有界带宽的平面图上 $r \geq 1$ 的 D$r$DSR 是 $\mathtt{PSPACE}$-完备的，改进了已知结果的度上界。我们还证明 $r=1$ 时二分图和弦图上在 $\mathsf{TS}$ 和 $\mathsf{TJ}$ 规则下已知的 $\mathtt{PSPACE}$-完备性结果可推广至 $r \geq 2$。