twin minus domination in directed graphs

let d=(v,a) be a finite simple directed graph. a function f:v⟶{−1,0,1} is called a twin minus dominating function if f(n−[v])≥1 and f(n+[v])≥1 for each vertex v∈v. the twin minus domination number of d is γ∗−(d)=min{w(f)∣f is a twin minus dominating function of d}. in this paper, we initiate the study of twin minus domination numbers in digraphs and present some lower bounds for γ∗−(d) in terms of the order, size and maximum and minimum in-degrees and out-degrees.