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.
Contributions to Discrete Mathematics 2(2) (2007)
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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.