Eternal Distance-2 Domination in Trees

TL;DR

This paper introduces a linear-time algorithm for the eternal distance-2 domination problem in trees, characterizes specific tree classes related to this problem, and provides bounds and constructions for eternal distance-k domination numbers.

Contribution

It presents the first linear-time algorithm for minimum eternal distance-2 domination in trees and characterizes trees with specific eternal domination properties.

Findings

Linear-time algorithm for eternal distance-2 domination in trees

Characterization of trees with equal eternal and standard domination numbers

Bounds and constructions for eternal distance-k domination numbers

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.