مشخصات پژوهش

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.