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Abstract
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Let k ≥ 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, . . . , k} such that for every v ∈ V (G) with f(v) = ∅, the condition
S
u∈NG(v)
f(u) = {1, 2, . . . , k} is fulfilled, where NG(v) is the open neighborhood of
v. The weight of a k-RDF f of G is the value ω(f) = P
v∈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 ∈ V (G) | f(v) 6= ∅} has
no isolated vertices. The total k-rainbow domination number of G, denoted by γtrk(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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