Global italian domination in graphs

An Italian dominating function (IDF) on a graph G = (V, E) is a function f: V ? {0, 1, 2} satisfying the condition that for every vertex v ? V (G) with f (v) = 0, either v is adjacent to a vertex assigned 2 under f, or v is adjacent to at least two vertices assigned 1. The weight of an IDF f is the value ?(v?V)((G)) f (v). The Italian domination number of a graph G, denoted by ?(I) (G), is the minimum weight of an IDF on G. An IDF f on G is called a global Italian dominating function (GIDF) on G if f is also an IDF on the complement G? of G. The global Italian domination number of G, denoted by ?(gI) (G), is the minimum weight of a GIDF on G. In this paper, we initiate the study of the global Italian domination number and we present some strict bounds for the global Italian domination number. In particular, we prove that for any tree T of order n ? 4, ?(gI) (T) ? ?(I) (T) + 2 and we characterize all trees with ?(gI) (T) = ?(I) (T) + 2 and ?(gI) (T) = ?(I) (T) + 1.