%0 Journal Article
%T On the outer independent 2-rainbow domination number of Cartesian products of paths and cycles
%J Communications in Combinatorics and Optimization
%I Azarbaijan Shahid Madani University
%Z 2538-2128
%A Dehgardi, Nasrin
%D 2021
%\ 12/01/2021
%V 6
%N 2
%P 315-324
%! On the outer independent 2-rainbow domination number of Cartesian products of paths and cycles
%K 2-rainbow dominating function
%K 2-rainbow domination number
%K outer independent 2-rainbow dominating function
%K outer independent 2-rainbow domination number
%K Cartesian product
%R 10.22049/cco.2021.27067.1188
%X 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 $vin V(G)$ with $f(v)=emptyset$, the condition $bigcup_{uin N_{G}(v)}f(u)={1,2}$ is fulfilled, where $N_{G}(v)$ is the open neighborhood of $v$. The weight of 2-RDF $f$ of $G$ is the value $omega (f):=sum _{vin V(G)}|f(v)|$. 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 $vin V(G)$ 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.
%U http://comb-opt.azaruniv.ac.ir/article_14166_529748fc23cb8af458c78bcc4dcc2d3a.pdf