Article
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Fault-Tolerant Detection Systems on the King's Grid
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Version 1
: Received: 16 January 2022 / Approved: 18 January 2022 / Online: 18 January 2022 (09:43:55 CET)
How to cite: Jean, D.; Seo, S. Fault-Tolerant Detection Systems on the King's Grid. Preprints 2022, 2022010249. https://doi.org/10.20944/preprints202201.0249.v1 Jean, D.; Seo, S. Fault-Tolerant Detection Systems on the King's Grid. Preprints 2022, 2022010249. https://doi.org/10.20944/preprints202201.0249.v1
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.
Keywords
domination, detection system, identifying-code, open-locating-dominating set, fault-tolerant, king's grid, density
Subject
Computer Science and Mathematics, Applied Mathematics
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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