While domination in (undirected) graphs is one of the most investigated topics in graph theory, domination in digraphs has been studied much less extensively. In this talk, we present some new results on (total) domination in digraphs with an emphasis on some digraph products. We present a generalization of the classical result of Meir and Moon from graphs to digraphs by proving that in an arbitrary ditree (a directed tree of which underlying graph is a tree) the domination number coincides with the packing number. In addition, a similar result is proved for the total domination number of a ditree. Then we focus on the total domination number of direct products of digraphs and the domination number of Cartesian products of digraphs. While the Vizing-type inequality is not true in all Cartesian products of digraphs, we present a different lower bound on the domination number of the Cartesian product of two digraphs expressed in terms of the domination numbers of factor digraphs, and demonstrate its sharpness.
Everyone is welcome and