Outer-Independent Italian Domination in Graphs

An outer-independent Italian dominating function (OIIDF) on a graph G with vertex set V(G) is defined as a function f : V(G) → {0, 1, 2}, such that every vertex v ∈ V(G) with f(v) = 0 has at least two neighbors assigned 1 under f or one neighbor w with f(w) = 2, and the set {u ∈ V | f (u) = 0} is independent. The weight of an OIIDF f is the value w(f) = Σ
_u∈V(G)
f(u). The minimum weight of an OIIDF on a graph G is called the outer-independent Italian domination number γ
_oiI
(G) of G. In this paper, we initiate the study of the outer-independent Italian domination number and present the bounds on the outer-independent Italian domination number in terms of the order, diameter, and vertex cover number. In addition, we establish the lower and upper bounds on γoiI (T) when T is a tree and characterize all extremal trees constructively. We also give the Nordhaus-Gaddum-type inequalities.