@article {
author = {Dehgardi, Nasrin},
title = {On the outer independent 2-rainbow domination number of Cartesian products of paths and cycles},
journal = {Communications in Combinatorics and Optimization},
volume = {6},
number = {2},
pages = {315-324},
year = {2021},
publisher = {Azarbaijan Shahid Madani University},
issn = {2538-2128},
eissn = {2538-2136},
doi = {10.22049/cco.2021.27067.1188},
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. },
keywords = {2-rainbow dominating function,2-rainbow domination number,outer independent 2-rainbow dominating function,outer independent 2-rainbow domination number,Cartesian product},
url = {http://comb-opt.azaruniv.ac.ir/article_14166.html},
eprint = {http://comb-opt.azaruniv.ac.ir/article_14166_529748fc23cb8af458c78bcc4dcc2d3a.pdf}
}