Graphs with Large Italian Domination Number

Abstract

An Italian dominating function on a graph G with vertex set V(G) is a function $$f :V(G) \rightarrow \{0,1,2\}$$ 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 $$n \ge 3$$ , then $$\gamma _{I}(G) \le \frac{3}{4}n$$ . Further, if G has minimum degree at least 2, then $$\gamma _{I}(G) \le \frac{2}{3}n$$ . 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.