A characterization of planar graphs by pseudo-line arrangements

Abstract

Let G 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 (G, E) is a graph for which the vertices are the pseudo-lines of G and the edges are some un-ordered pairs of pseudo-lines of G. A diamond of pseudo-line graph (G, E) is a pair of edges {p, q}, {pʹ, qʹ} ϵ E, {p, q} Ç {pʹ, qʹ} = ø, such that the crossing point of the pseudo-lines p and q lies vertically between pʹ and qʹ and the crossing point of pʹ and qʹ 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(kl/3n) upper bound on the k-level complexity of an arrangement of straightfines, which is very recently discovered by Dey, holds for an arrangement of pseudo-lines as well.