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