Pair Crossing Number, Cutwidth, and Good Drawings on Arbitrary Point Sets

Abstract

Determining whether there exists a graph such that its crossing number and pair crossing number are distinct is an important open problem in geometric graph theory. We show that cr ( G ) = O ( pcr ( G ) 3 / 2 ) for every graph G, improving the previous best bound by a logarithmic factor. Answering a question of Pach and Tóth, we prove that the bisection width (and, in fact, the cutwidth as well) of a graph G with degree sequence d 1 , d 2 , ⋯ , d n satisfies bw ( G ) = O ( pcr ( G ) + ∑ k = 1 n d k 2 ) . Then we show that there is a constant C ≥ 1 such that the following holds: For any graph G of order n and any set S of at least n C points in general position on the plane, G admits a straight-line drawing which maps the vertices to points of S and has no more than O log n · pcr ( G ) + ∑ k = 1 n d k 2 crossings. Our proofs rely on a slightly modified version of a separator theorem for string graphs by Lee, which might be of independent interest.