Generalized power domination in regular graphs

In this paper, we continue the study of power domination in graphs (see [T. W. Haynes et al., SIAM J. Discrete Math., 15 (2002), pp. 519-529; P. Dorbec et al., SIAM J. Discrete Math., 22 (2008), pp. 554-567; A. Aazami et al., SIAM J. Discrete Math., 23 (2009), pp. 1382-1399]). Power domination in graphs was birthed from the problem of monitoring an electric power system by placing as few measurement devices in the system as possible. A set of vertices is defined to be a power dominating set of a graph if every vertex and every edge in the system is monitored by the set following a set of rules (according to Kirschoff laws) for power system monitoring. The minimum cardinality of a power dominating set of a graph is its power domination number. We show that the power domination of a connected cubic graph on n vertices different from K3,3 is at most n/4 and this bound is tight. More generally, we show that for k ≥ 1, the k-power domination number of a connected (k+2)-regular graph on n vertices different from K_k+2,k+2 is at most n/(k+3), where the 1-power domination number is the ordinary power domination number. We show that these bounds are tight.