Visibility Number of Directed Graphs
Published in SIAM Journal on Discrete Mathematics, 2013
M. Axenovich, A. Beveridge, J. P. Hutchinson and D. West, Visibility Number of Directed Graphs, SIAM Journal on Discrete Mathematics, Vol 27, No. 3, (2013), 1429-1449.
Preprint link: https://github.com/mathbeveridge/mathbeveridge.github.io/blob/master/files/dvnum.pdf
A $k$-bar visibility representation of a digraph $$G assigns each vertex at most $k$ horizontal segments in the plane so that $G$ has an arc $uv$ if and only if some segment for $u$ "sees" some segment for $v$ above it by a vertical line of sight. The (bar) visibility number $b(G)$ of a digraph $G$ is the least $k$ permitting such a representation. We prove bounds on the bar visibility of planar digraphs and tournaments.