Italian domination and perfect Italian domination on Sierpiński graphs

Abstract

An Italian dominating function (IDF) of a graph G is a function f : V(G) → {0, 1, 2} satisfying the condition that for every v ∈ V with f(v) = 0, Σ u ∈N(v) f(u) ≥ 2. The weight of an IDF on G is the sum f(V) = Σ v ∈V f(v) and the Italian domination number, γ I (G), is the minimum weight of an IDF. An IDF is a perfect Italian dominating function (PID) on G, if for every vertex v ∈ V(G) with f(v) = 0 the total weight assigned by f to the neighbours of v is exactly 2, i.e., all the neighbours of u are assigned the weight 0 by f except for exactly one vertex v for which f(v) = 2 or for exactly two vertices v and w for which f(v) = f(w) = 1. The weight of a PID-function is f(V) = Σ u ∈V(G) f(u). The perfect Italian domination number of G, denoted by , is the minimum weight of a PID-function of G. In this paper, we obtain the Italian domination number and perfect Italian domination number of Sierpiński graphs.