Stijn Cambie
Published 2022-01-01, updated 2022-04-18Version 2
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.
Comments: 22 pages, 15 figures, this paper is an extended version of a second part of arXiv:1903.01358, mainly clarifying the content of section 5, where the bipartite cases are considered as well. Furthermore a foreword and table with figures of the extremal (di)graphs has been added for clarification
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