Failed power domination in grids, cylinders, and tori

Byrne, Isabel T.; Matthews, Gretchen L.; Patel, Nikita M.; Trivedi, Anuradha G.; Winslow, Margaret A.

doi:10.2140/involve.2023.16.705

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Failed power domination in grids, cylinders, and tori

Isabel T. Byrne, Gretchen L. Matthews, Nikita M. Patel, Anuradha G. Trivedi and Margaret A. Winslow

Vol. 16 (2023), No. 4, 705–717

DOI: 10.2140/involve.2023.16.705

Abstract

The power domination number of a graph is the minimum number of vertices required to monitor the graph. Here, the notion of monitoring is given by a set of rules for power system monitoring where vertices model phasor management units (PMU) in a power network. We consider the failed power domination number of a graph $G$ , $γ_{fp} (G)$ , a recently introduced graph parameter. Any set of vertices of $G$ whose cardinality is greater than $γ_{fp} (G)$ will dominate the graph. The failed power domination number also allows one to consider PMU (or node) failure. Indeed, any set of $γ_{fp} (G) + i + 1$ vertices will monitor the network even in the presence of $i$ node failures. We establish the failed power domination number for products of paths and cycles including square grids, tori, and hypercubes and provide bounds for the failed power domination number of square cylinders.

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Keywords

failed power domination, graph product, grid, power domination

Mathematical Subject Classification

Primary: 05C15, 05C78

Secondary: 05C50

Milestones

Received: 20 July 2022

Revised: 19 October 2022

Accepted: 27 October 2022

Published: 31 October 2023

Communicated by Glenn Hurlbert

Authors

Isabel T. Byrne
Department of Mathematical Sciences University of Delaware Newark, DE United States

Gretchen L. Matthews
Department of Mathematics Virginia Tech Blacksburg, VA United States

Nikita M. Patel
Department of Computer Science Virginia Tech Blacksburg, VA United States

Anuradha G. Trivedi
Department of Biomedical Engineering Georgia Tech and Emory University Atlanta, GA United States

Margaret A. Winslow
Department of Mathematics Virginia Tech Blacksburg, VA United States

Vol. 16, No. 4, 2023