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

TL;DR

This work studies fault-tolerant locating-dominating sets on the infinite king grid, focusing on three variants: RED:LD, DET:LD, and ERR:LD. It develops share-based and discharging techniques to derive lower bounds and provides explicit tilings to establish upper bounds, yielding precise density intervals for each variant: $\mathrm{RED:LD}(\mathrm{K})\in[\tfrac{3}{11},\tfrac{5}{16}]$, $\mathrm{DET:LD}(\mathrm{K})\in[\tfrac{3}{11},\tfrac{3}{8}]$, and $\mathrm{ERR:LD}(\mathrm{K})\in[\tfrac{15}{38},\tfrac{7}{16}]$. The methods advance fault-tolerant LD theory on dense infinite grids and provide constructions and techniques (share, discharging) applicable to related graphs.

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}.

Figures (20)

• Figure 1: Example RED:LD set.
• Figure 2: Our best constructions of LD (a), RED:LD (b), DET:LD (c), and ERR:LD (d) sets on the King's grid. Shaded vertices denote detectors.
• Figure 3: ERR:LD set to demonstrate Theorem \ref{['theo:err-ld-char']}
• Figure 4: Configurations around a detector vertex $x$.
• Figure 5: L shape

Theorems & Definitions (20)

• Theorem 1.1: jean23b
• Theorem 1.2: jean23b
• Theorem 1.3: jean23a
• Theorem 1.4: errld
• Corollary 1.1: jean23b
• Corollary 1.2: errld
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Definition 3.4