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 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. 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.

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