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Abstract
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Let k ≥ 1 be an integer and G be a simple graph with vertex set V (G). Let f be a
function that assigns label from the set {0, 1, 2, . . . , k + 1} to the vertices of a graph
G. For a vertex v ∈ V (G), the active neighborhood of v, denoted by AN(v), is the
set of vertices w ∈ NG(v) such that f(w) ≥ 1. A [k]-RDF is a function f : V (G) →
{0, 1, 2, . . . , k + 1} satisfying the condition that for any vertex v ∈ V (G) with f(v) < k,
f(NG[v]) ≥ |AN(v)| + k. The weight of a [k]-RDF is ω(f) = Σv∈V (G)f(v). The [k]-
Roman domination number γ[kR](G) of G is the minimum weight of an [k]-RDF on
G. The case k = 4 is called quadruple Roman domination number. In this paper, we
first establish an upper bound for quadruple Roman domination number of graphs with
minimum degree two, and then we derive a Nordhaus–Gaddum bound on the quadruple
Roman domination number of graphs.
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