The global Roman domination number of torus graphs and generalized Petersen graphs

摘要

Let γ_(gR)(C_m□C_n) denote the global Roman domination number of the torus graph formed by the Cartesian product of the graphs C_m, the path of length m, m ≥ 3 and the graph C_n, the cycle of length n, n ≥ 3. In this paper, we determine the exact values of generalized Petersen graphs P(n, 2) and P{n, 3) and γ_(gR)(C_m□C_n) for m = 3,4,5. Moreover, we provide the exact values for some torus graphs γ_(gR)(C_m□C_n) with smal-1 m and n. We conclude that γ_(gR)(C_m□C_n) = γ_R(C_m□C_n) for any m ≥ n ≥ 3 and γ_(gR)(P(n,k)) = γ_R(P{n,k)) for any n ≥ 2k + 1 ≥ 9.