A characterization of planar graphs by pseudo-line arrangements

Let Γ be an arrangement of pseudo-lines, i.e., a collection of unbounded x-monotone curves in which each curve crosses each of the others exactly once. A pseudo-line graph (Γ, E) is a graph for which the vertices are the pseudo-lines of Γ and the edges are some unordered pairs of pseudo-lines of Γ. A diamond of a pseudo-line graph (Γ, E) is a pair of edges {p, q}, {p′, q′} ∈ E, {p, q} ∩ {p′, q′} = Ø, such that the crossing point of the pseudo-lines p and q lies vertically between p′ and q′ and the crossing point of p′ and q′ lies vertically between p and q. We show that a graph is planar if and only if it is isomorphic to a diamond-free pseudo-line graph. An immediate consequence of this theorem is that the O(k^1/3 n) upper bound on the k-level complexity of an arrangement of straight lines, which was very recently discovered by Dey, holds for an arrangement of pseudo-lines as well.