Authors: Paul Dorbec ¹; Antonio González ^2,3,4; Claire Pennarun ⁵

• 1 Université de Caen Normandie
• 2 Department of Didactics of Mathematics, Faculty of Educational Sciences, Universidad de Sevilla
• 3 Universidad de Sevilla / University of Sevilla
• 4 Universidad de Sevilla = University of Seville
• 5 Laboratoire Bordelais de Recherche en Informatique

Power domination in graphs emerged from the problem of monitoring an electrical system by placing as few measurement devices in the system as possible. It corresponds to a variant of domination that includes the possibility of propagation. For measurement devices placed on a set S of vertices of a graph G, the set of monitored vertices is initially the set S together with all its neighbors. Then iteratively, whenever some monitored vertex v has a single neighbor u not yet monitored, u gets monitored. A set S is said to be a power dominating set of the graph G if all vertices of G eventually are monitored. The power domination number of a graph is the minimum size of a power dominating set. In this paper, we prove that any maximal planar graph of order n ≥ 6 admits a power dominating set of size at most (n−2)/4 .

• 05C10 - Planar graphs; geometric and topological aspects of graph theory
• 05C35 - Extremal problems in graph theory
• 05C69 - Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
• 05C90 - Applications of graph theory