Generalized power domination in regular graphs
Résumé
In this paper, we continue the study of power domination in graphs (see SIAM J. Discrete Math. 15 (2002), 519-529; SIAM J. Discrete Math. 22 (2008), 554-567; SIAM J. Discrete Math. 23 (2009), 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 Kk+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.
Domaines
Mathématique discrète [cs.DM]
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