New results on quadruple Roman domination in graphs.

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 ∈ N G (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 (N G [ v ]) ≥ | AN (v) | + k. The weight of a [ k ] -RDF is ω (f) = Σ v ∈ V (G) f (v). The [ k ] -Roman domination number γ [ k R ] (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. [ABSTRACT FROM AUTHOR]