On 3-Rainbow Domination Number of Silicate Network

Abstract

Given an undirected graph $G=(V, E)$ and a set of k-colors numbered 1,2{\dots},k. The 3-rainbow domination defined as $f : V(G)\to \mathcal{├╛}\{1,2,3\}$ such that for each vertex $v \in V(G)$ with $f(v)=\emptyset$. We have, \[\bigcup_{u\in N(v)}{f\left(u\right)=\{1,2,3\}}\] Such a function f is called a 3-rainbow dominating function (3RDF) and minimum weight of such function is called the 3-rainbow domination number of G and is denoted by ${\gamma}_{r3}(G)$. In this paper we find the 3-rainbow domination number of interconnection network.