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Abstract
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Let G be a finite and simple graph with vertex set V (G). Let f be a function that assigns
label from the set {0, 1, 2, 3, 4, 5} 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 quadruple Roman dominating function (QRDF) is a function
f : V (G) −→ {0, 1, 2, 3, 4, 5} satisfying the condition that for any vertex v ∈ V (G) with
f(v) < 4, f(NG[v]) ≥ |AN(v)| + 4. The weight of a QRDF is ω(f) = Σv∈V (G)f(v). The
quadruple Roman domination number γ[4R](G) of G is the minimum weight of a QRDF
on G. In this paper, we investigate the properties of the quadruple Roman domination
number of graphs, present bounds on γ[4R](G) and give exact values for some graph
families. In addition, complexity results are also obtained.
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