  Visibility representation of digraphs was introduced by Axenovich, Beveridge,
Hutch\-inson, and West (\emph{SIAM J. Discrete Math.} {\bf 27}(3) (2013)
1429--1449) as a natural generalization of $t$-bar visibility representation of
undirected graphs. A {\it $t$-bar visibility representation} of a digraph $G$
assigns each vertex at most $t$ horizontal bars in the plane so that there is
an arc $xy$ in the digraph if and only if some bar for $x$ "sees" some bar for
$y$ above it along an unblocked vertical strip with positive width. The {\it
visibility number} $b(G)$ is the least $t$ such that $G$ has a $t$-bar
visibility representation. In this paper, we solve several problems about
$b(G)$ posed by Axenovich et al.\ and prove that determining whether the bar
visibility number of a digraph is $2$ is NP-complete.
