# Total $k$-Rainbow domination numbers in graphs

Document Type : Original paper

Authors

1 Babol Noshirvani University of Technology

3 Shahrood University of Technology

4 University of Architecture, Civil Engineering and Geodesy

Abstract

Let $kgeq 1$ be an integer, and let $G$ be a graph. A {it
$k$-rainbow dominating function} (or a {it $k$-RDF}) of $G$ is a
function $f$ from the vertex set $V(G)$ to the family of all subsets
of ${1,2,ldots ,k}$ such that for every $vin V(G)$ with
$f(v)=emptyset$, the condition $bigcup_{uin N_{G}(v)}f(u)={1,2,ldots,k}$ is fulfilled, where $N_{G}(v)$ is
the open neighborhood of $v$. The {it weight} of a $k$-RDF $f$ of
$G$ is the value $omega (f)=sum _{vin V(G)}|f(v)|$. A $k$-rainbow
dominating function $f$ in a graph with no isolated vertex is called
a {em total $k$-rainbow dominating function} if the subgraph of $G$
induced by the set ${v in V(G) mid f (v) not = {color{blue}emptyset}}$ has no isolated vertices. The {em total $k$-rainbow domination number} of $G$, denoted by
$gamma_{trk}(G)$, is the minimum weight of a total $k$-rainbow
dominating function on $G$. The total $1$-rainbow domination is the
same as the total domination. In this paper we initiate the
study of total $k$-rainbow domination number and we investigate its
basic properties. In particular, we present some sharp bounds on the
total $k$-rainbow domination number and we determine {color{blue}the} total
$k$-rainbow domination number of some classes of graphs.

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Main Subjects