On the Power Dominating Sets of Hypercubes

The performance of electrical networks is monitored by expensive Phasor Measurement Units (PMUs). It is economically beneficial to determine the optimal placement and the minimum number of PMUs required to effectively monitor an entire network. This problem has a graph theory model involving power dominating sets in a graph. A set S of vertices in a graph is called a power dominating set if every vertex and every edge in the graph is "observed" by S according to a set of observation rules. The power domination number of a graph is the minimum cardinality of a power dominating set of the graph. In this paper, the power domination number is determined for hypercubes Q_n with n = 2^k, where k is any positive integer.