When does a digraph have infinite radius?

Question

For a graph with infinite radius, the graph is disconnected. When does a digraph have infinite radius? Is it possible for a connected digraph to have infinite radius?

Answer 1

Let $G$ have vertex set $\Bbb Z$ and edges $\{n,n+1\}$ for $n\in\Bbb Z$, so that $G$ is a bi-infinite chain; then for each $n\in\Bbb Z$ the eccentricity of $n$ is $\sup\{|k-n|:k\in\Bbb Z\setminus\{n\}\}=\infty$, so $G$ has infinite radius. $G$ is, however, connected and bipartite: the parts are the sets of even and odd integers.