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Power domination in maximal planar graphs
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
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