On the power domination number of de Bruijn and Kautz digraphs

Let G = (V,A) be a directed graph without parallel arcs, and let S ⊆ V be a set of vertices. Let the sequence S = S0 ⊆ S1 ⊆ S2 ⊆ · · · be defined as follows: S1 is obtained from S0 by adding all out-neighbors of vertices in S0. For k > 2, Sk is obtained from Sk−1 by adding all vertices w such that for some vertex v ∈ Sk−1, w is the unique out-neighbor of v in V \ Sk−1. We set M(S) = S0 ∪S1 ∪ · · · , and call S a power dominating set for G if M(S) = V (G). The minimum cardinality of such a set is called the power domination number of G. In this paper, we determine the power domination numbers of de Bruijn and Kautz digraphs.