# On the outer independent 2-rainbow domination number of Cartesian products of paths and cycles

Document Type : Original paper

Author

Sirjan University of Technology, Sirjan 78137, Iran

Abstract

Let $G$ be a graph. A  $2$-rainbow dominating function (or {\em 2-RDF}) of $G$ is a function $f$ from $V(G)$ to the set of all subsets of the set $\{1,2\}$ such that for a vertex $v\in V(G)$ with $f(v)=\emptyset$, the condition $\bigcup_{u\in N_{G}(v)}f(u)=\{1,2\}$ is fulfilled, where $N_{G}(v)$ is the open neighborhood of $v$. The weight of 2-RDF $f$ of $G$ is the value $\omega (f):=\sum _{v\in V(G)}|f(v)|$. The {\em $2$-rainbow domination number} of $G$, denoted by $\gamma_{r2}(G)$, is the minimum weight of a 2-RDF of $G$. A 2-RDF $f$ is called an  outer independent $2$-rainbow dominating function (or  OI2-RDF} of $G$ if the set of all $v\in V(G)$ with $f(v)=\emptyset$ is an independent set. The outer independent $2$-rainbow domination number $\gamma_{oir2}(G)$ is the minimum weight of an OI2-RDF of $G$. In this paper, we obtain the outer independent $2$-rainbow domination number of $P_{m}\square P_{n}$ and $P_{m}\square C_{n}$. Also we determine the value of $\gamma_{oir2}(C_{m}\Box C_{n})$ when $m$ or $n$ is even.

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