Visibility Number of Directed Graphs

Abstract

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. Among other results, we show that $b(G) \leq 4$ when $G$ is a planar digraph (reducing to 3 when the underlying graph has no triangles), $b(G) \leq 2$ when $G$ is outerplanar, and $b(G) \leq (n+10)/3$ when $G$ has $n$ vertices. When $G$ is the $n$-vertex transitive tournament, $b(G) \leq 7n/24 + 2\sqrt{n \log n}$, improving to $b(G) < 3n/14 + 42$ when $n$ is sufficiently large. Our tools include arboricity, interval number, and Steiner systems.