The bar visibility number of a graph

The bar visibility number of a graph G, denoted b(G), is the minimum t such that G can be represented by assigning each vertex x the set S-x of points in at most t horizontal segments in the plane so that uv ∈ E(G) if and only if some point of S-u sees some point of S-v via a vertical segment of positive width unobstructed by assigned points. Among our results are the following: (1) Every planar graph has bar visibility number at most 2, which is sharp. (2) r ≤ b(K-m,K-n) ≤ r + 1, where r = [mn+4/2m+2n]. (3) b(K-n) = [n/6]. (4) If G has n vertices, then b(G) ≤ [n/6] + 2.