Inducing regulation of any digraphs

For a given structure DD (digraph, multidigraph, or pseudodigraph) and an integer rr large enough, a smallest inducing rr-regularization of DD is constructed. This regularization is an rr-regular superstructure of the smallest possible order with bounded arc multiplicity, and containing DD as an induced substructure. The sharp upper bound on the number, ρρ, of necessary new vertices among such superstructures for nn-vertex general digraphs DD is determined, ρρ being called the inducing regulation number of DD. For Δ̃(D) being the maximum among semi-degrees in DD, simple nn-vertex digraphs DD with largest possible ρρ are characterized if either r≥Δ̃(D) or r=Δ̃(D) (where the case r=Δ̃ is not a trivial subcase of r≥Δ̃).