Maximum size of digraphs of given radius

In 1967, Vizing determined the maximum size of a graph with given order and radius. In 1973, Fridman answered the same question for digraphs with given order and outradius. We investigate that question when restricting to strong digraphs. Strong digraphs are the digraphs with a finite total distance and hence the interesting ones, as we want to note a connection between minimizing the total distance and maximizing the size under the same constraints. We characterize the extremal digraphs maximizing the size among all strong digraphs of order n and outradius 3, as well as when the order is sufficiently large compared to the outradius. As such, we solve a problem of Dankelmann asymptotically. We also consider these questions for bipartite digraphs and solve a second problem of Dankelmann partially. The full paper can also be found at arxiv.org/abs/2201.00186.(c) 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).