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On Power Domination of Generalized Petersen Graphs

The power dominating problem is a variation of the classical domination problem in graphs. Electricity company use phase measurement units (PMUs) to produce the measuring data of a system, and use these data to estimate states of the system. Because of the high cost of PMUs, minimizing the number of PMUs on a system is an important problem for electricity companies. This problem can be modeled by graphs as follows: Given an undirected graph ( , ) G V E = , find a set ( ) S V G ⊆ with minimum cardinality such that all vertices in V are observed by some vertices in S. Herein, a vertex which is claimed to be a PMU observes itself and all its neighbors, and if an observed vertex has all but one of its neighbors observed, then the remaining neighbor becomes observed as well. The minimum size of S for graph G is called power dominating number and is denoted as ( ) p G γ . This paper provides some tight upper bounds on power domination number of generalized Petersen graphs.