Maximum size of digraphs of given radius

Abstract

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 biconnected digraphs. Biconnected 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 biconnected digraphs of order n and outradius 3, as well as when the order is sufficiently large compared to the outradius. By doing so, we solve a problem of Dankelmann asymptotically. We also consider these questions for bipartite digraphs.