# 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 $k\geq 1$ be an integer, and let $G$ be a graph. A $k$-rainbow dominating function (or a {$k$-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 $v\in V(G)$ with $f(v)=\emptyset$, the condition $\bigcup_{u\in N_{G}(v)}f(u)=\{1,2,\ldots,k\}$ is fulfilled, where $N_{G}(v)$ is the open neighborhood of $v$. The  weight of a $k$-RDF $f$ of $G$ is the value $\omega (f)=\sum _{v\in V(G)}|f(v)|$. A $k$-rainbow dominating function $f$ in a graph with no isolated vertex is called a  total $k$-rainbow dominating function if the subgraph of $G$ induced by the set $\{v\in V(G) \mid f (v) \neq \emptyset\}$ has no isolated vertices. The  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 the total $k$-rainbow domination number of some classes of graphs.

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