Reconfiguration problems ask whether one feasible solution can be transformed into another by a sequence of local moves while maintaining feasibility throughout. For integers $d \geq 1$ and $k \geq d+1$, the Distance Coloring problem asks if a given graph $G$ has a $(d, k)$-coloring, i.e., a coloring of the vertices of $G$ by $k$ colors such that any two vertices within distance $d$ from each other have different colors. For ordinary proper colorings ($d=1$), the $k$-Coloring Reconfiguration problem is polynomial-time solvable for $k\le 3$ [Cereceda, van den Heuvel, and Johnson, J. Graph Theory 67(1):69--82, 2011] but is $\mathsf{PSPACE}$-complete for every fixed $k\ge 4$, even on bipartite graphs [Bonsma and Cereceda, Theor. Comput. Sci. 410(50):5215--5226, 2009]. In this work, we initiate a study of the distance-$d$ analogue, for $d \geq 2$. We show that even for planar, bipartite, and $2$-degenerate graphs, $(d, k)$-Coloring Reconfiguration remains $\mathsf{PSPACE}$-complete for every $d \geq 3$ via a reduction from the well-known Sliding Tokens problem. Our construction uses $k = k_0 + 2 + n(\lceil d/2\rceil-1)$ colors on instances of size $n$, where $k_0\in\{3d+3,3d+6\}$ (depending on the parity of $d$). For $d = 2$, the same reduction scheme can be adapted to show that the problem is $\mathsf{PSPACE}$-complete on planar and $2$-degenerate graphs with same values of $k$. Additionally, on split graphs, there is an interesting dichotomy: the problem is $\mathsf{PSPACE}$-complete when $d = 2$ and $k$ is large but can be solved efficiently when $d \geq 3$ and $k \geq d+1$. For chordal graphs, we show that the problem is $\mathsf{PSPACE}$-complete for even values of $d \geq 2$. Finally, we design a quadratic-time algorithm to solve the problem on paths for any $d \geq 2$ and $k \geq d+1$.

翻译：重配置问题探讨是否可以通过一系列局部移动，在始终保持可行性的前提下，将一个可行解变换为另一个可行解。对于整数 $d \geq 1$ 和 $k \geq d+1$，距离着色问题询问给定图 $G$ 是否存在一个 $(d, k)$-着色，即使用 $k$ 种颜色对 $G$ 的顶点进行着色，使得任意两个距离不超过 $d$ 的顶点颜色不同。对于普通正常着色（$d=1$），$k$-着色重配置问题在 $k\le 3$ 时可在多项式时间内求解 [Cereceda, van den Heuvel, and Johnson, J. Graph Theory 67(1):69--82, 2011]，但对于每个固定的 $k\ge 4$，即使在二分图上也是 $\mathsf{PSPACE}$-完全的 [Bonsma and Cereceda, Theor. Comput. Sci. 410(50):5215--5226, 2009]。本文中，我们首次研究 $d \geq 2$ 时的距离-$d$ 模拟问题。我们证明，即使对于平面、二分且 $2$-退化图，$(d, k)$-着色重配置问题对于每个 $d \geq 3$ 仍然是 $\mathsf{PSPACE}$-完全的，其归约来自著名的滑动令牌问题。对于规模为 $n$ 的实例，我们的构造使用 $k = k_0 + 2 + n(\lceil d/2\rceil-1)$ 种颜色，其中 $k_0\in\{3d+3,3d+6\}$（取决于 $d$ 的奇偶性）。对于 $d = 2$，可调整相同的归约方案，证明该问题在平面且 $2$-退化图（使用相同的 $k$ 值）上仍是 $\mathsf{PSPACE}$-完全的。此外，在分裂图上存在一个有趣的分叉：当 $d = 2$ 且 $k$ 较大时，问题是 $\mathsf{PSPACE}$-完全的，但当 $d \geq 3$ 且 $k \geq d+1$ 时可高效求解。对于弦图，我们证明该问题对于偶数 $d \geq 2$ 是 $\mathsf{PSPACE}$-完全的。最后，我们设计了一个二次时间算法，用于解决路径上任意 $d \geq 2$ 且 $k \geq d+1$ 的情况。