Power domination in block graphs

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 domination problem in graphs. In 2002, Haynes et al. considered the graph theoretical representation of this problem as a variation of the domination problem. They defined 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 systemmonitoring). The power domination number p(G) of a graph G is the minimum cardinality of a power dominating set of G. This problem was proved NP-complete even when restricted to bipartite graphs and chordal graphs. In this paper, we present a linear time algorithm for solving the power domination problem in block graphs. As an application of the algorithm, we establish a sharp upper bound for power domination number in block graphs and characterize the extremal graphs. © 2006 Elsevier B.V. All rights reserved.