Domination in a digraph and in its reverse

Let DD be a digraph. By γ(D)γ(D) we denote the domintaion number of DD and by D−D− we denote a digraph obtained by reversing all the arcs of DD. In this paper we prove that for every δ≥3δ≥3 and k≥1k≥1 there exists a simple strongly connected δδ-regular digraph Dδ,kDδ,k such that γ(Dδ,k−)−γ(Dδ,k)=k. Analogous theorem is obtained for total domination number provided that δ≥4δ≥4.