Generalized Power Domination: Propagation Radius and Sierpiński Graphs

Abstract

The recently introduced concept of k-power domination generalizes domination and power domination, the latter concept being used for monitoring an electric power system. The k-power domination problem is to determine a minimum size vertex subset S of a graph G such that after setting X = N [S], and iteratively adding to X vertices x that have a neighbour v in X such that at most k neighbours of v are not yet in X, we get X = V (G). In this paper the k-power domination number of Sierpiński graphs is determined. The propagation radius is introduced as a measure of the efficiency of power dominating sets. The propagation radius of Sierpiński graphs is obtained in most of the cases.

Grundy dominating sequences and zero forcing sets

“…Being in a given state, a propagation step consist of coloring all vertices blue that may be colored blue in the state. Analogous concept for the power domination (under the name propagation radius) was introduced in [14] and independently in [24].…”

Graphs that are simultaneously efficient open domination and efficient closed domination graphs

“…The Sierpiński graphs S n p were introduced in [26] and afterwards investigated from many different aspects. Here we only mention recent studies of Sierpiński graphs related to codes and domination [11,17,37], their shortest paths [22,43], and an appealing Assume now that p is even, say p = 2k, k ≥ 2. We first recall from [27] that S n p contains an ECD set.…”

Power domination throttling

“…See, e.g., [6,10,13,20,21,29,42,44] for various structural and computational results about power domination and related variants. The power propagation time of a graph has previously been studied in [1,19,24,31]. Other variants of propagation time arising from similar dynamic graph coloring processes have also been studied; these include zero forcing propagation time [7,23,27,28] and positive semidefinite propagation time [40].…”