Total $k$-Rainbow domination numbers in graphs

Abstract

Let $kgeq 1$ be an integer, and let $G$ be a graph. A $k$-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 subsetsof ${1,2,ldots ,k}$ 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 weight of a $k$-RDF $f$ of$G$ is the value $omega (f)=sum _{vin V(G)}|f(v)|$. A $k$-rainbowdominating function $f$ in a graph with no isolated vertex is calleda total $k$-rainbow dominating function if the subgraph of $G$induced by the set ${vin V(G) mid f (v) neq emptyset}$ has no isolated vertices. The total $k$-rainbow domination number of $G$, denoted by$gamma_{trk}(G)$, is the minimum weight of a total $k$-rainbowdominating function on $G$. The total $1$-rainbow domination is thesame as the total domination. In this paper we initiate thestudy of total $k$-rainbow domination number and we investigate itsbasic properties. In particular, we present some sharp bounds on thetotal $k$-rainbow domination number and we determine the total$k$-rainbow domination number of some classes of graphs.