Maximum size of digraphs of given radius

TL;DR

This paper investigates the maximum size of biconnected digraphs with a given order and outradius, characterizing extremal structures and connecting size maximization with total distance minimization.

Contribution

It characterizes extremal biconnected digraphs maximizing size for outradius 3 and large order, extending previous results to a more restricted class of digraphs.

Findings

Characterization of extremal digraphs for outradius 3

Asymptotic solution to Dankelmann's problem

Partial results for bipartite digraphs

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. 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.