Fault-Tolerant Detection Systems on the King's Grid

TL;DR

This paper studies fault-tolerant detection systems on the infinite king's grid, constructing optimal redundant sets that ensure unique intruder detection even if one detector fails, and establishes bounds on their minimum densities.

Contribution

It introduces fault-tolerant variants of OLD and IC detection systems, providing constructions and bounds for their minimum densities on the king's grid.

Findings

Bounds on minimum densities: [3/10, 1/3] for RED:OLD

Bounds on minimum densities: [3/11, 1/3] for RED:IC

Constructed optimal fault-tolerant detection sets

Abstract

A detection system, modeled in a graph, uses "detectors" on a subset of vertices to uniquely identify an "intruder" at any vertex. We consider two types of detection systems: open-locating-dominating (OLD) sets and identifying codes (ICs). An OLD set gives each vertex a unique, non-empty open neighborhood of detectors, while an IC provides a unique, non-empty closed neighborhood of detectors. We explore their fault-tolerant variants: redundant OLD (RED:OLD) sets and redundant ICs (RED:ICs), which ensure that removing/disabling at most one detector guarantees the properties of OLD sets and ICs, respectively. This paper focuses on constructing optimal RED:OLD sets and RED:ICs on the infinite king's grid, and presents the proof for the bounds on their minimum densities; [3/10, 1/3] for RED:OLD sets and [3/11, 1/3] for RED:ICs.