Some new results on bar visibility of digraphs

TL;DR

This paper investigates the visibility number of directed graphs, providing solutions to open problems and proving that deciding if this number equals two is computationally hard (NP-complete).

Contribution

It solves several open problems about the visibility number and establishes NP-completeness for a key decision problem.

Findings

Solved multiple open problems on the visibility number.

Proved NP-completeness of deciding if the visibility number is 2.

Extended understanding of visibility representations of digraphs.

Abstract

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.