距离着色
Distance Recoloring
AI总结：
研究探讨了距离着色重配置问题，证明对于d≥3，平面图、双色图等的(d,k)-着色重配置问题均为PSPACE难，且在路径图上设计了二次时间算法。
摘要：
重配置问题询问是否可通过局部移动将一个可行解转换为另一个，保持可行性。对于整数d≥1和k≥d+1，距离着色问题询问给定图G是否存在(d,k)-着色，即用k种颜色对G的顶点着色，使得任意两个距离不超过d的顶点颜色不同。对于普通proper着色(d=1)，k-着色重配置问题在k≤3时多项式时间可解，但对k≥4固定时为PSPACE难，即使在双色图上。本文研究距离-d类比，d≥2。证明即使在平面图、双色图和2-退化图上，(d,k)-着色重配置问题对d≥3仍为PSPACE难，通过滑动令牌问题归约。构造使用k=k0+2+n(⌈d/2⌉-1)种颜色，k0∈{3d+3,3d+6}（根据d的奇偶性）。对于d=2，相同归约方案可显示在平面图和2-退化图上问题仍为PSPACE难。此外，在分割图上存在有趣二元性：当d=2且k较大时问题为PSPACE难，但当d≥3且k≥d+1时可高效解决。对于弦图，证明当d≥2为偶数时问题为PSPACE难。最后，设计二次时间算法解决路径图上的问题，对于任何d≥2和k≥d+1。
英文摘要：
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 $k \geq d+1$.
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