Fault-tolerant Locating-Dominating Sets on the Infinite King Grid

TL;DR

This paper investigates fault-tolerant locating-dominating sets in the infinite king grid, establishing bounds for their minimum densities and enhancing network fault detection capabilities.

Contribution

It introduces and analyzes three types of fault-tolerant LD sets, providing bounds for their minimum densities in the infinite king grid.

Findings

Bounds for minimum densities of fault-tolerant LD sets established

Analysis of redundant, error-detecting, and error-correcting LD sets

Enhanced fault detection in network models

Abstract

Let $G$ be a graph of a network system with vertices, $V(G)$, representing physical locations and edges, $E(G)$, representing informational connectivity. A \emph{locating-dominating (LD)} set $S \subseteq V(G)$ is a subset of vertices representing detectors capable of sensing an "intruder" at precisely their location or somewhere in their open-neighborhood -- an LD set must be capable of locating an intruder anywhere in the graph. We explore three types of fault-tolerant LD sets: \emph{redundant LD} sets, which allow a detector to be removed, \emph{error-detecting LD} sets, which allow at most one false negative, and \emph{error-correcting LD} sets, which allow at most one error (false positive or negative). In particular, we determine lower and upper bounds for the minimum density of these three fault-tolerant locating-dominating sets in the \emph{infinite king grid}.