Some new results on bar visibility of digraphs

• DOI: 10.1016/j.dam.2024.09.024
• Publication Date: 2024-10-01T14:36:32Z
• Authors: Yuanrui Feng; Jun Ge; Douglas B. West; Yan Yang
• Keywords: FOS: Mathematics; Mathematics - Combinatorics; Combinatorics (math.CO); 0102 computer and information sciences; 05C62, 05C35, 05C10, 68Q17; 01 natural sciences

17 pages,9 figures<br/>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.<br/>