On the total Italian domination number in digraphs

Consider a finite simple digraph D with vertex set V (D). An Italian dominating function (IDF) on D is a function f: V (D) \0, 1, 2\ satisfying every vertex u with f (u) =0 has an in-neighbor v with f (v) =2 or two in-neighbors w and z with f (w) =f (z) =1. A total Italian dominating function (TIDF) on D is an IDF f such that the subdigraph D\ u\, |\, f (u) 1\ contains no isolated vertices. The weight (f) of a TIDF f on D is ₔ ₕ (₃) f (u). The total Italian domination number of D is ₓ₈ (D) =\ (f) \, |\, f is a TIDF on D\. In this paper, we present bounds on ₓ₈ (D), and investigate the relationship between several different domination parameters. In particular, we give the total Italian domination number of the Cartesian products P₂ Pₙ and P₃ Pₙ, where Pₙ represents a dipath with n vertices.