DOMINATION IN GRAPHS APPLIED TO ELECTRIC POWER NETWORKS ∗ TERESA W. HAYNES † , SANDRA M. HEDETNIEMI ‡ , STEPHEN T. HEDETNIEMI ‡ , AND MICHAEL A. HENNING § SIAM J. DISCRETE MATH. c  2002 Society for Industrial and Applied Mathematics Vol. 15, No. 4, pp. 519–529 Abstract. The problem of monitoring an electric power system by placing as few measurement devices in the system as possible is closely related to the well-known vertex covering and dominating set problems in graphs. We consider the graph theoretical representation of this problem as a variation of the dominating set problem and define a set S to be a power dominating set of a graph if every vertex and every edge in the system 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 G is the power domination number γ P (G). We show that the power dominating set (PDS) problem is NP-complete even when restricted to bipartite graphs or chordal graphs. On the other hand, we give a linear algorithm to solve the PDS for trees. In addition, we investigate theoretical properties of γ P (T ) in trees T . Key words. domination, power domination, electric power monitoring AMS subject classification. 05C69 PII. S0895480100375831 1. Introduction. Electric power companies need to continually monitor their system’s state as defined by a set of state variables (for example, the voltage magnitude at loads and the machine phase angle at generators [6]). One method of monitoring these variables is to place phase measurement units (PMUs) at selected locations in the system. Because of the high cost of a PMU, it is desirable to minimize their number while maintaining the ability to monitor (observe) the entire system. A system is said to be observed if all of the state variables of the system can be determined from a set of measurements (e.g., voltages and currents). Let G =(V,E) be a graph representing an electric power system, where a vertex represents an electrical node (a substation bus where transmission lines, loads, and generators are connected) and an edge represents a transmission line joining two electrical nodes. The problem of locating a smallest set of PMUs to monitor the entire system is a graph theory problem closely related to the well-known vertex covering and domination problems. Hence, this problem is not only of interest in the power system industry but also as a new problem in graph theory. For a thorough study of domination and related subset problems as well as terminology not defined here, we refer the reader to two books [4, 5]. A PMU measures the state variable (voltage and phase angle) for the vertex at which it is placed and its incident edges and their endvertices. (These vertices and edges are said to be observed.) The other observation rules are as follows: * Received by the editors July 24, 2000; accepted for publication (in revised form) June 20, 2002; published electronically September 10, 2002. This research was supported in part by the South African National Research Foundation and the University of Natal, South Africa. http://www.siam.org/journals/sidma/15-4/37583.html † Department of Mathematics, East Tennessee State University, Johnson City, TN 37614 (haynes@ etsu.edu). ‡ Department of Computer Science, Clemson University, Clemson, SC 29634 (shedet@cs.clemson. edu, hedet@cs.clemson.edu). § Department of Mathematics, University of Natal, Private Bag X01, Pietermaritzburg, 3209 South Africa (henning@math.unp.ac.za). 519