On the Rainbow Domination Number of Digraphs

Abstract

For a positive integer k, a k-rainbow dominating function (kRDF) of a digraph D is a function f from the vertex set V(D) to the set of all subsets of $\{1,2,\ldots ,k\}$ such that for any vertex v with $f(v)=\emptyset$, $\bigcup _{u\in N^-_D(v)}f(u)=\{1,2,\ldots ,k\}$, where $N^-_D(v)$ is the set of in-neighbors of v. The weight of a kRDF f is defined as $\sum _{v\in V(D)}|f(v)|$. The k-rainbow domination number of a digraph D, denoted by $\gamma _{rk}(D)$, is the minimum weight of a kRDF of D. We establish several bounds on $\gamma _{rk}(D)$. Moreover, we determine the exact value of $\gamma _{rk}(D)$ in Cartesian product of directed cycles for $k\ge 4$.