Let $G$ be a directed graph.


Call a vertex $v$ in $G$ *central* if there exists $\Theta(n^2)$ distinct pairs of vertices $(u,w)$ such that $v$ lies on some path from $u$ to $w$. We do not care whether these paths are shortest or not, nor whether other paths avoiding $v$ exist.


Question: Do strongly connected digraphs always admit a central vertex?



Does a strong digraph always admit a vertex that lies on some path between $\Theta(n^2)$ pairs of vertices?
In [this manuscript](https://arxiv.org/abs/2304.03567), Bessy, Thomassé, and Viennot proved the following stronger property:


Let $D$ be a strongly connected directed graph, then there is a vertex $v$ in $D$ such that there exists an in-tree towards $v$ and an out-tree from $v$ that are vertex-disjoint (apart from sharing $v$ itself) and contain at least $n/6$ vertices each.


As a result, $v$ lies on a path between at least $n^2/36$ pairs of vertices.


