مشخصات پژوهش

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.