Visibility Number of Directed Graphs

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) ≤ 4 when G is a planar digraph (reducing to 3 when the underlying graph has no triangles), b(G) ≤ 2 when G is outerplanar, and b(G) ≤ (n + 10)/3 when G has n vertices. When G is the n-vertex transitive tournament, b(G) ≤ 7n/24 + 2 √ n log n, improving to b(G) < 3n/14 + 42 when n is sufficiently large. Our tools include arboricity, interval number, and Steiner systems.