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arXiv · 2308.00054

Eternal Distance-2 Domination in Trees

Alexander Clow·Christopher M van Bommel
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Abstract

We consider the eternal distance-2 domination problem, recently proposed by Cox, Meger, and Messinger, on trees. We show that finding a minimum eternal distance-2 dominating set of a tree is linear time in the order of the graph by providing a fast algorithm. Additionally, we characterise when trees have an eternal distance-2 domination number equal to their domination number or their distance-2 domination number, along with characterizing which trees are eternal distance-2 domination critical. We conclude by providing general upper and lower bounds for the eternal distance-k domination number of a graph, as well as constructing an infinite family of trees which meet said upper bound and another which meets the given lower bound.

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Alexander Clow, Christopher M van Bommel. 2023-07-31. Eternal Distance-2 Domination in Trees. https://arxiv.org/abs/2308.00054

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