## Abstract

Let G be a simple graph with vertex set V and edge set E, and let S ⊆ V . The sets Pi (S), i ≥ 0, of vertices monitored by S at the i th step are given by P0(S) = N[S] and Pi+1(S) = Pi (S) {w : {w} = N[v]\Pi (S) for some v ∈ Pi (S)}. If there exists j such that Pj (S) = V , then S is called a power dominating set, PDS, of G. Otherwise, S is a failed power dominating set, FPDS. The power domination number of a simple graph G, denoted γp(G) gives the minimum number of measurement devices known as phasor measurement units (PMUs) required to observe a power network represented by G, and is the minimum cardinality of any PDS of G. The failed power domination number of G, ¯γp(G), is the maximum cardinality of any FPDS of G, and represents the maximum number of PMUs that could be placed on a given power network represented by G, but fail to observe the full network. As a consequence, ¯γp(G)+1 gives the minimum number of PMUs necessary to successfully observe the full network no matter where they are placed. We prove that ¯γp(G) is NP-hard to compute, determine graphs in which every vertex is a PDS, and compare ¯γp(G) to similar parameters.

## Creative Commons License

This work is licensed under a Creative Commons Attribution-No Derivative Works 4.0 International License.

## Publication Date

2-1-2020

## Document Type

Article

## Department, Program, or Center

Computer Science (GCCIS)

## Recommended Citation

Glasser, Abraham; Jacob, Bonnie; Lederman, Emily; and Radziszowski, Stanislaw, "Failed power domination on graphs" (2020). *Australasian Journal of Combinatorics*, 76 (2), 232-247. Accessed from

https://repository.rit.edu/article/1982

## Campus

RIT – Main Campus