Some new results on bar visibility of digraphs

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.