The m-eternal domination problem and its Roman and Italian variants are NP-hard on narrow graph classes, and four infinite regular grids admit strongly optimal eternally-dominating guard sets.
Journal of Combinatorial Mathematics and Combinatorial Computing50, 179–194 (2004)
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m-Eternal Domination and Variants on Some Classes of Finite and Infinite Graphs
The m-eternal domination problem and its Roman and Italian variants are NP-hard on narrow graph classes, and four infinite regular grids admit strongly optimal eternally-dominating guard sets.