﻿ Total \$k\$-Rainbow domination numbers in graphs

Total \$k\$-Rainbow domination numbers in graphs

چکیده

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 afunction \$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_{uinN_{G}(v)}f(u)={1,2,ldots,k}\$ is fulfilled, where \$N_{G}(v)\$ isthe 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\$-rainbowdominating function \$f\$ in a graph with no isolated vertex is calleda {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\$-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 {color{blue}the} total\$k\$-rainbow domination number of some classes of graphs.

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عنوان ژورنال:

دوره 3  شماره 1

صفحات  37- 50

تاریخ انتشار 2018-06-01

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