Geodesic
Power domination in maximal planar graphs
Discrete mathematics & theoretical computer science, Tome 21 (2019) no. 4.

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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 .
DOI : 10.23638/DMTCS-21-4-18
Classification : 05C10, 05C35, 05C69, 05C90 
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Dorbec, Paul; González, Antonio; Pennarun, Claire. Power domination in maximal planar graphs. Discrete mathematics & theoretical computer science, Tome 21 (2019) no. 4. doi : 10.23638/DMTCS-21-4-18. http://geodesic.mathdoc.fr/articles/10.23638/DMTCS-21-4-18/

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