Eternal Domination in Trees

TL;DR

This paper studies the minimum number of mobile guards needed to defend trees against infinite attacks, providing characterizations of trees that meet specific bounds on this number.

Contribution

It offers new characterizations of trees that achieve bounds on the $m$-eternal domination number, advancing understanding of guarding strategies in graph theory.

Findings

Characterized trees achieving upper bounds on $m$-eternal domination number.

Characterized trees achieving lower bounds on $m$-eternal domination number.

Provided insights into guard placement strategies in trees.

Abstract

Mobile guards on the vertices of a graph are used to defend the graph against an infinite sequence of attacks on vertices. A guard must move from a neighboring vertex to an attacked vertex (we assume attacks happen only at vertices containing no guard). More than one guard is allowed to move in response to an attack. The $m$-eternal domination number is the minimum number of guards needed to defend the graph. We characterize the trees achieving several upper and lower bounds on the $m$-eternal domination number.