The complexity of minimum-density locating-dominating set in infinite periodic graphs

2026-08-03

AI Summary

This study investigates the computational complexity of finding minimum-density locating-dominating sets (LDS) in infinite ℤ-periodic graphs. By constructing a periodic graph model and designing a combinatorial reduction, the authors extend an NP-hard problem from finite graphs to the infinite setting, thereby establishing—for the first time—that the minimum-density LDS problem remains NP-hard on infinite periodic graphs. This work bridges a theoretical gap between cardinality minimization in finite graphs and density minimization in infinite graphs, while also providing a transferable framework for analyzing the computational complexity of optimization problems over a broad class of infinite structures.

Abstract

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.

Problem

• locating-dominating set
• minimum density
• infinite periodic graphs
• NP-hardness
• computational complexity

Innovation

• locating-dominating set
• infinite periodic graphs
• minimum density
• NP-hardness
• computational complexity