Eternal Domination: D-Dimensional Cartesian and Strong Grids and Everything in Between

Abstract

In the eternal domination game played on graphs, an attacker attacks a vertex at each turn and a team of guards must move a guard to the attacked vertex to defend it. The guards may only move to adjacent vertices on their turn. The goal is to determine the eternal domination number γ ∞ all of a graph, which is the minimum number of guards required to defend against an infinite sequence of attacks. This paper first continues the study of the eternal domination game on strong grids P n P m. Cartesian grids P n P m have been vastly studied with tight bounds existing for small grids such as k ×n grids for k ∈ {2, 3, 4, 5}. It was recently proven that γ ∞ all (P n P m) = γ(P n P m) + O(n + m) where γ(P n P m) is the domination number of P n P m which lower bounds the eternal domination number [Lamprou et al. Eternally dominating large grids. Theoretical Computer Science, 794:27-46, 2019]. We prove that, for all n, m ∈ N * such that m ≥ n, n 3 m 3 + Ω(n + m) = γ ∞ all (P n P m) = n 3 m 3 + O(m √ n) (note that n 3 m 3 is the domination number of P n P m). We then generalise our technique to prove that γ ∞ all (G) = γ(G) + o(γ(G)) for all graphs G ∈ F, where F is a large family of D-dimensional grids which are supergraphs of the D-dimensional Cartesian grid and subgraphs of the Ddimensional strong grid. In particular, F includes both the D-dimensional Cartesian grid and the D-dimensional strong grid.

Computing m-Eternal Domination Number of Cactus Graphs in Linear Time

“…Since then, several results have focused on finding bounds on γ ∞ m under different conditions or graph classes. Among the studied graph classes are trees [12,8,14], grids [4,18,16,10,5,9,15], and interval graphs [2,17]. For a good survey of other related results and topics, see Klostermeyer and Mynhardt [13].…”