The Rainbow Domination Number of a Digraph


J. Amjadi, A. Bahremandpour, S. M. Sheikholeslami, L. Volkmann




Let $D=(V,A)$ be a finite and simple digraph. A II-rainbow dominating function} (2RDF) of a digraph $D$ is a function $f$ from the vertex set $V$ to the set of all subsets of the set $\{1,2\}$ such that for any vertex $v\in V$ with $f(v)=\emptyset$ the condition $\bigcup_{u\in N^-(v)}f(u)=\{1,2\}$ is fulfilled, where $N^-(v)$ is the set of in-neighbors of $v$. The weight of a 2RDF $f$ is the value $\omega(f)=\sum_{v\in V}|f (v)|$. The $2$-rainbow domination number of a digraph $D$, denoted by $\gamma_{r2}(D)$, is the minimum weight of a 2RDF of $D$. In this paper we initiate the study of rainbow domination in digraphs and we present some sharp bounds for $\gamma_{r2}(D)$.