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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

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, γfp (G), a recently introduced graph parameter. Any set of vertices of G whose cardinality is greater than γfp (G) will dominate the graph. The failed power domination number also allows one to consider PMU (or node) failure. Indeed, any set of γfp (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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failed power domination, graph product, grid, power domination
Mathematical Subject Classification
Primary: 05C15, 05C78
Secondary: 05C50
Received: 20 July 2022
Revised: 19 October 2022
Accepted: 27 October 2022
Published: 31 October 2023

Communicated by Glenn Hurlbert
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