Eternal Domination Numbers of $5\times n$ Grid Graphs

Abstract

Eternal domination of a graph requires the positioning of guards to protect against an infinitely long sequence of attacks where, in response to an attack, each guard can either remain in place or move to a neighbouring vertex, while keeping the graph dominated. This paper investigates the $m$-eternal domination numbers for $5\times n$ grid graphs. The values, previously known for $1\le n\le 5$, are determined for $6\le n\le 12$, and lower and upper bounds derived for $n>12$.