Power Domination in Graphs.

Abstract

The problem of monitoring an electric power system by placing phase measuring units can be studied as a power dominating set problem in graph theory. This problem is closely related to the well-known domination problem in graphs and is a close cousin of the well-studied zero-forcing set. A set S of vertices is defined to be a power dominating set of a graph if every vertex and every edge in the graph is monitored by the set S, following a set of rules for power system monitoring. The minimum cardinality of a power dominating set of a graph is its power domination number. In this thesis, we present upper bounds on the power domination numbers for general Cartesian products of two graphs, as well as Cartesian products of three graphs from the same classes of graphs. We also present a global upper bound for cubic graphs and determine the power domination number for circulant graphs. Finally, the power domination numbers for Mycielski graphs are characterized.

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