Some new results on bar visibility of digraphs

Abstract

Visibility representation of digraphs was introduced by Axenovich et al. (2013) as a natural generalization of $t$-bar visibility representation of undirected graphs. A $t$-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 visibility number $b\left(G\right)$ is the least $t$ such that $G$ has a $t$-bar visibility representation. In this paper, we solve several problems about $b\left(G\right)$ posed by Axenovich et al. and prove that determining whether the bar visibility number of a digraph is 2 is NP-complete.