MMN-1267
Global rainbow domination in graphs
J. Amjadi; S. M. Sheikholeslami; L. Volkmann;Abstract
For a positive integer $k$, a {\em $k$-rainbow dominating
function} (kRDF) of a graph $G$ is a function $f$ from the vertex
set $V(G)$ to the set of all subsets of the set $\{1,2,\ldots,k\}$
such that for any vertex $v\in V(G)$ with $f(v)=\emptyset$ the
condition $\bigcup_{u\in N(v)}f(u)=\{1,2,\ldots,k\}$ is fulfilled,
where $N(v)$ is the neighborhood of $v$. The {\em weight} of a
kRDF $f$ is the value $\omega(f)=\sum_{v\in V}|f (v)|$. A kRDF $f$
is called a {\em global $k$-rainbow dominating function} (GkRDF)
if $f$ is also a kRDF of the complement $\overline{G}$ of $G$.
The {\em global $k$-rainbow domination number} of $G$, denoted by
$\gamma_{grk}(G)$, is the minimum weight of a GkRDF on $G$. In
this paper, we initiate the study of global $k$-rainbow domination
numbers and we establish some sharp bounds for it.
Vol. 17 (2016), No. 2, pp. 749-759