On Two Outer Independent Roman Domination Related Parameters in Torus Graphs

Abstract

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 (γoidR(G)) γoiR(G) is the minimum weight taken over all (OIDRDFs) OIRDFs of G. In this article, we investigate these two parameters γoiR(G) and γoidR(G) of regular graphs and present lower bounds on them. We improve the lower bound on γoiR(G) for a regular graph presented by Ahangar et al. (2017). Furthermore, we present upper bounds on γoiR(G) and γoidR(G) for torus graphs. Furthermore, we determine the exact values of γoiR(C3□Cn) and γoiR(Cm□Cn) for m≡0(mod4) and n≡0(mod4), and the exact value of γoidR(C3□Cn). By our result, γoidR(Cm□Cn)≤5mn/4 which verifies the open question is correct for Cm□Cn that was presented by Ahangar et al. (2020).