AI 中文总结
本文针对矩形网格图，通过递归构造和闭式公式，完整刻画并枚举了所有可能基数（含偶数及奇数）的最小分辨集，解决了开放问题。
AI 中文摘要
设 $m,n \geq 3$。矩形网格图 $P_m \square P_n$ 的顶点集 $S$ 称为分辨集，如果 $P_m \square P_n$ 的顶点关于 $S$ 的出租车距离向量两两不同。分辨集 $S$ 称为最小的，如果 $S$ 的任何真子集都不是分辨集；基数为 $k$ 的最小分辨集称为 $k$-最小集。我们扩展了 Andersen 等人的工作，他们刻画了 $3$-最小集，确定了最大基数 $2\min(m,n)-2$，并将网格图最小分辨集的完整刻画与枚举作为开放问题提出；我们还扩展了 Adar 和 Epstein 的工作，他们证明了 $3$ 是唯一可能的奇数基数，且每个基数至少为 $4$ 的最小分辨集都可以排序形成对应于锯齿序列的序列。我们提供了一种递归构造，恰好生成网格图中基数至少为 $4$ 的最小分辨集。我们由该构造推导出闭式公式，用于枚举所有偶数 $4 \leq k \leq 2\min(m,n) -2$ 的 $k$-最小集。结合已知的基数 $2$ 和 $3$ 的刻画以及 $3$-最小集的直接枚举，这给出了矩形网格图所有可能基数的最小分辨集的完整刻画与枚举。
英文摘要
Let $m,n \geq 3$. A set of vertices $S$ of the rectangular grid graph $P_m \square P_n$ is resolving if the taxicab distance vectors of the vertices of $P_m \square P_n$ with respect to $S$ are pairwise distinct. A resolving set $S$ is minimal if no proper subset of $S$ is resolving, and a minimal resolving set of cardinality $k$ is called a $k$-minimal. We extend the work of Andersen et al., who characterized $3$-minimals, established the maximum cardinality $2\min(m,n)-2$, and posed the complete characterization and enumeration of minimal resolving sets for grids as an open problem, and Adar and Epstein, who showed that $3$ is the only possible odd cardinality and that every minimal resolving set of cardinality at least $4$ can be ordered to form a sequence corresponding to a zigzag sequence. We provide a recursive construction that generates exactly the minimal resolving sets of cardinality at least $4$ for grids. We derive from the construction closed-form formulas to enumerate the $k$-minimals for all even $4 \leq k \leq 2\min(m,n) -2$. Together with the known characterizations for cardinalities $2$ and $3$ and a direct enumeration of the $3$-minimals, this yields a complete characterization and enumeration of the minimal resolving sets of rectangular grid graphs of every possible cardinality.
Comments29 pages, 4 figures
