Co-Roman dominating of girds

Let G = (V, E) be a simple graph with vertex set 𝑉 and let 𝑓: 𝑉 → {0,1,2} be a function of weight 𝜔(𝑓) = ∑ 𝑓(𝑣) 𝑣∈𝑉(𝐺) . A vertex 𝑣 is protected with respect to 𝑓, if 𝑓(𝑣) > 0 or 𝑓(𝑣) = 0 and 𝑣 is adjacent to a vertex 𝑢 such that 𝑓(𝑢) > 0. The function 𝑓 is a co-Roman dominating function, abbreviated CRDF if: (i) every vertex 𝑢 with 𝑓(𝑢) = 0 is adjacent to a vertex 𝑣 for which 𝑓(𝑣) > 0, and (ii) every vertex 𝑣 with 𝑓(𝑣) > 0 has a neighbor 𝑢 for which 𝑓(𝑢) = 0, such that each vertex of 𝐺 is protected with respect to the function 𝑓 ′ : 𝑉(𝐺) → {0,1,2}, defined by 𝑓 ′ (𝑣) = 𝑓(𝑣) −1, 𝑓 ′ (𝑢) = 1 and 𝑓 ′ (𝑥) = 𝑓(𝑥) for 𝑥 ∈ 𝑉(𝐺) − {𝑢, 𝑣}. The co-Roman domination number of a graph G, denoted by 𝛾𝑐𝑟(𝐺), is the minimum weight of a co-Roman dominating function on G. In this paper, we study the co-Roman domination number of grid graphs and we obtain this parameter for 𝑃2 × 𝑃𝑛 and 𝑃3 × 𝑃𝑛.