Power domination problem in graphs

To monitor an electric power system by placing as few phase measurement units (PMUs) as possible is closely related to the famous vertex cover problem and domination problem in graph theory. A set S is a power dominating set (PDS) of a graph G = (V, E), if every vertex and every edge in the system is observed following the observation rules of power system monitoring. The minimum cardinality of a PDS of a graph G is the power domination number gamma(p)(G). We show that the problem of finding the power domination number for split graphs, a subclass of chordal graphs, is NP-complete. In addition, we present a linear time algorithm for finding gamma(p)(G) of an interval graph G, if the interval ordering of the graph is provided, and show that the algorithm with O(n log n) time complexity, is asymptotically optimal, if the interval ordering is not given, where n is the number of intervals. We also show that the same results hold for the class of proper circular-arc graphs.