On Two Outer Independent Roman Domination Related Parameters in Torus Graphs

In a graph G=(V,E), where every vertex is assigned 0, 1 or 2, f an assignment such that 0 has at least one neighbor 2 and all vertices labeled by are independent, then called outer independent Roman dominating function (OIRDF). The domination strengthened if 1, 3, each two neighbors double (OIDRDF). weight of (OIDRDF) OIRDF the sum f(v) for v?V. (double) number (?oidR(G)) ?oiR(G) minimum taken over (OIDRDFs) OIRDFs G. this article, we investigate these parameters ?oidR(G) regular graphs present lower bounds on them. We improve bound presented Ahangar et al. (2017). Furthermore, upper torus graphs. determine exact values ?oiR(C3?Cn) ?oiR(Cm?Cn) m?0(mod4) n?0(mod4), value ?oidR(C3?Cn). By our result, ?oidR(Cm?Cn)?5mn/4 which verifies open question correct Cm?Cn was (2020).