AI 中文总结
该研究证明具有有限周期的无限ℤ-周期图中最小密度定位支配集问题是NP难题，通过周期归约弥合了有限图与无限图相关问题的复杂性研究缺口。
AI 中文摘要
图G的支配集S若满足：对每个不在S中的不同顶点对，它们在S中的邻域互不相同，则S称为定位支配集（LDS）。在有限图中寻找最小基数的LDS是著名的NP难题。在无限图中，该问题自然推广为寻找最小密度的LDS。尽管针对特定无限正则网格的密度界限已被广泛研究，但无限图的计算复杂性结果仍属空白。我们证明，具有有限周期的无限ℤ-周期图中的最小密度LDS问题是NP难题。该结果通过严格的周期归约，弥合了有限图的基数最小化与无限图的密度最小化之间的差距。此外，我们的方法可用于建立无限周期图上相关结构问题的NP难解性。
英文摘要
A dominating set $S$ of a graph $G$ is a locating-dominating set (LDS) if, for each pair of distinct vertices not in~$S$, their neighbourhoods in $S$ are distinct. Finding a minimum-cardinality LDS in finite graphs is a well-known NP-hard problem. On infinite graphs, this problem naturally generalises to finding an LDS of minimum density. While density bounds have been widely studied for specific infinite regular grids, no computational complexity results exist for infinite graphs. We prove that the minimum-density LDS problem in infinite $\mathbb{Z}$-periodic graphs with a finite period is NP-hard. This result bridges the gap between cardinality minimization on finite graphs and density minimization on infinite graphs via a rigorous periodic reduction. Furthermore, our approach can be adapted to establish NP-hardness for related structural problems on infinite periodic graphs.
Comments10 pages with 4 figures
