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 γ_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 γI(G)≤34n. Further, if G has minimum degree at least 2, then γI(G)≤23n. 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.