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We show sharp Vizing-type inequalities for eternal domination. Namely, we prove that for any graphs G and H, where is the eternal domination function, α is the independence number, and is the strong product of graphs. This addresses a question of Klostermeyer and Mynhardt. We also show some families of graphs attaining the strict inequality where is the Cartesian product. For the eviction model of eternal domination, we show a sharp upper bound for