Nordhaus–Gaddum type inequalities on the total Italian domination number in graphs

Abstract

Let G be a graph with vertex set V(G). A total Italian dominating function (TIDF) on a graph G is a function f : V(G) → {0, 1, 2} such that (i) every vertex v with f(v) = 0 is adjacent to a vertex u with f(u) = 2 or to two vertices w and z with f(w) = f(z) = 1, and (ii) every vertex v with f(v) ≥ 1 is adjacent to a vertex u with f(u) ≥ 1. The total Italian domination number γtI(G) on a graph G is the minimum weight of a total Italian dominating function. In this paper, we present Nordhaus–Gaddum type inequalities for the total Italian domination number.