Open-locating-dominating sets with error correction

TL;DR

This paper introduces and analyzes error-correcting open-locating-dominating sets in graphs, providing complexity results, existence criteria, and exploring extremal and infinite grid cases for fault-tolerant detection systems.

Contribution

It defines a fault-tolerant variant of open-locating-dominating sets, proves NP-completeness, and characterizes existence conditions for arbitrary graphs.

Findings

NP-completeness of the minimum error-correcting open-locating-dominating set problem

Characterization of existence criteria for such sets in arbitrary graphs

Analysis of extremal graphs and infinite grid cases

Abstract

An open-locating-dominating set of a graph models a detection system for a facility with a possible "intruder" or a multiprocessor network with a possible malfunctioning processor. A "sensor" or "detector" is assumed to be installed at a subset of vertices where it can detect an intruder or a malfunctioning processor in their neighborhood, but not at itself. We consider a fault-tolerant variant of an open-locating-dominating set called an error-correcting open-locating-dominating set, which can correct a false-positive or a false-negative signal from a detector. In particular, we prove the problem of finding a minimum error-correcting open-locating-dominating set in an arbitrary graph is NP-complete. Additionally, we characterize the existence criteria for an error-correcting open-locating-dominating sets for an arbitrary graph. We also consider extremal graphs that require every vertex to be a detector and minimum error-correcting open-locating-dominating sets in infinite grids.