Consider a graph G of order n and maximum degree ∆. Let f:V(G)→{0,1,…,⌈Δ2⌉+1} be a function that labels the vertices of G. Let B₀ = {v ∈ V (G) : f(v) = 0}. The function f is a strong Roman dominating function for G if every v ∈ B₀ has a neighbor w such that f(w)≥1+⌈12|N(w)∩B0|⌉. In this paper, we study the bounds on strong Roman domination numbers of the Cartesian product P_m □P_k of paths P_m and paths P_k. We compute the exact values for the strong Roman domination number of the Cartesian product P₂ □P_k and P₃ □P_k. We also show that the strong Roman domination number of the Cartesian product P₄ □P_k is between ⌈13(8k−⌊k8⌋+1⌉ and ⌈8k3⌉ for k ≥ 8, and that both bounds are sharp bounds.

Keywords: Roman domination number, strong Roman domination number, grid.