Graphs with Large Italian Domination Number

An Italian dominating function on a graph G with vertex set V(G) is a function f : V(G) -> {0, 1, 2} having the property that for every vertex v with f (v) = 0, at least two neighbors of v are assigned 1 under f or at least one neighbor of v is assigned 2 under f. The weight of an Italian dominating function f is the sum of the values assigned to all the vertices under f. The Italian domination number of G, denoted by gamma(I)(G), is the minimum weight of an Italian dominating of G. It is known that if G is a connected graph of order n >= 3, then gamma(I)(G) <= 3/4n. Further, if G has minimum degree at least 2, then gamma(I) (G) <= 2/3n. In this paper, we characterize the connected graphs achieving equality in these bounds. In addition, we prove Nordhaus-Gaddum inequalities for the Italian domination number.