Variations on the power domination problem: Hypergraphs and robustness
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The power domination problem seeks to find the minimum number of sensors called phasor measurement units (PMUs) to monitor an electric power network. In this dissertation, we present two variations of the power domination problem.
The first variation is infectious power domination, which is a new way to generalize the power domination problem to hypergraphs using the infection rule from Bergen et al. (2018). We compare to the previous generalization by Chang and Roussel (2015). We examine general bounds; graph families such as complete k-partite hypergraphs, circular arc hypergraphs, and trees; and the impact of edge/vertex removal, linear sums, Cartesian products, and weak coronas.
The second variation considers how the minimum number of sensors and their placement changes when k sensors are allowed to fail. The PMU-defect robust power domination number is also a novel parameter, generalizing the work done by Pai, Chang, and Wang (2010) by allowing multiple sensors to be placed at the same location. We give general bounds, explicit values for some complete bipartite graphs, and computational results for small square grid graphs. We also give a new proof of the power domination number for trees and conjecture the PMU-defect robust power domination number for trees.