This question is an attempt to make progress on domotorp's interesting challenge. I'll first state the version of this which I would most expect to already be studied; then I'll state the version which would be enough for my purposes.

Let $G$ be a directed graph on $n$ vertices. For any vertices $u$ and $v$, let $\delta(u,v)$ be the length of the shortest directed path from $u$ to $v$ and let $d(u,v)$ be the length of the shortest path from $u$ to $v$ ignoring edge orientations. We will assume that $\delta(u,v)$ (and hence $d(u,v)$) if finite; the term for this is that the graph is strongly connected. I'll write $d=\max_{(u,v)} d(u,v)$ and $\delta = \max_{(u,v)} \delta(u,v)$.

Is there a bound for $\delta$ in terms of $d$, independent of $n$?

That's the one I'd expect to have already been studied by someone else. Here is what I actually need: Define a directed graph to have the pairwise domination property if, for any two distinct vertices $u$ and $v$ of $G$, there is a vertex $x$ with $u \rightarrow x \leftarrow v$. (In particular, this implies $d \leq 2$.) What I really need is:

Is there an integer $k$ such that every strongly connected graph on $> 2$ vertices with the pairwise domination property contains an oriented $k$-cycle?

A positive answer to the first question implies one for the second: If there is a directed path $u \leadsto v$ and a directed path $v \leadsto u$ both of length $\leq \delta$, then the union of these paths contains a directed cycle of length $\leq 2 \delta$.

A positive answer to this question shows that the parameter $\epsilon$, in domotrop's question, can't be taken less than $1/k$.