## Abstract

Let*G* be a graph of*n* vertices that can be drawn in the plane by straight-line segments so that no*k*+1 of them are pairwise crossing. We show that*G* has at most*c*_{ k }*n*log^{2k−2}*n* edges. This gives a partial answer to a dual version of a well-known problem of Avital-Hanani, Erdós, Kupitz, Perles, and others. We also construct two point sets {*p*_{1},⋯,*p*_{ n }}, {*q*_{1},⋯,*q*_{ n }} in the plane such that any piecewise linear one-to-one mapping*f*∶**R**^{2}→**R**^{2} with*f(pi)*=*qi* (1≤*i*≤*n*) is composed of at least Ω(*n*^{2}) linear pieces. It follows from a recent result of Souvaine and Wenger that this bound is asymptotically tight. Both proofs are based on a relation between the crossing number and the bisection width of a graph.

## Key words

Crossing number Geometric graph Bisection width Triangulation## Preview

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