# A characterization of planar graphs by pseudo-line arrangements

## Abstract

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* (*ΓT, E*) is a graph for which the vertices are the pseudo-lines of *Γ* and the edges are some un-ordered pairs of pseudo-lines of *Γ*. A *diamond* of pseudo-line graph (*Γ, E*) is a pair of edges *p, q*), *p′, q′*) ∈ *E*, (*p′, q′*) ∩ *p′, q′* = 0, 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*^{⅓}*n*) upper bound on the *k*-level complexity of an arrangement of straight-lines, which is very recently discovered by Dey, holds for an arrangement of pseudo-lines as well.

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