Outer Independent Roman Domination Number of Cartesian Product of Paths and Cycles

TL;DR

This paper investigates the outer independent Roman domination number for Cartesian products of paths and cycles, providing exact values for small cases and bounds for larger graphs.

Contribution

It determines exact values of the outer independent Roman domination number for certain small Cartesian product graphs and offers an upper bound for larger cases.

Findings

Exact values for $n=1,2,3$ and $m=3$ cases.

Upper bounds for larger $n$ and $m$.

Methodology for calculating domination numbers in product graphs.

Abstract

Given a graph $G$ with vertex set $V$, an outer independent Roman dominating function (OIRDF) is a function $f$ from $V(G)$ to $\{0, 1, 2\}$ for which every vertex with label $0$ under $f$ is adjacent to at least a vertex with label $2$ but not adjacent to another vertex with label $0$. The weight of an OIRDF $f$ is the sum of vertex function values all over the graph, and the minimum of an OIRDF is the outer independent Roman domination number of $G$, denoted as $\gamma_{oiR}(G)$. In this paper, we focus on the outer independent Roman domination number of the Cartesian product of paths and cycles $P_{n}\Box C_{m}$. We determine the exact values of $\gamma_{oiR}(P_n\Box C_m)$ for $n=1,2,3$ and $\gamma_{oiR}(P_n\Box C_3)$ and present an upper bound of $\gamma_{oiR}(P_n\Box C_m)$ for $n\ge 4, m\ge 4$.