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arXiv · 1706.10047

Power domination in maximal planar graphs

Paul Dorbec·Antonio González·Claire Pennarun
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Abstract

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 $\ge$ 6 admits a power dominating set of size at most (n--2)/4 .

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Paul Dorbec, Antonio González, Claire Pennarun. 2019-12-11. Power domination in maximal planar graphs. https://arxiv.org/abs/1706.10047

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