Inducing regulation of any digraphs Original Research Article

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