Counting Power Domination Sets in Complete m-ary Trees

Abstract

The study of power domination sets arises from the monitoring of electrical network using Phase Measurement Units (PMUs or monitors). This problem was first studied in terms of graphs in [4] in 2002 and has been a topic of much interest since then (see e.g. [1–3, 6, 7]). A PMU placed at a network node measures the voltage at the node and all current phasors at the node [1], and subsequently measures the voltage at some neighboring nodes using the propagation rules described in Definition 1. Since PMUs are expensive, it is desirable to find the minimum number of PMUs needed to monitor a network. This problem is known to be to be NP-complete even for planar bipartite graphs ([3]). Since the cost of technology typically decreases but the cost of employment increases, it is feasible that the cost of placing extra PMUs is preferred to the cost of determining the minimum number of PMUs and an optimal placement. Thus, in this paper, we begin to investigate how probable it is that a randomly placed set of k PMUs will monitor a network.