Pair Crossing Number, Cutwidth, and Good Drawings on Arbitrary Point Sets
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454_2024_Article_708.pdf
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336.97 KB
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82573225e066e8921921f03dd0eb0b63
Author(s)
Pi, Oriol S.
Date Issued
January 22, 2025
Journal
Discrete & Computational Geometry
Publisher
Springer US
Citation
Pi, O.S. Pair Crossing Number, Cutwidth, and Good Drawings on Arbitrary Point Sets. Discrete Comput Geom 73, 310–326 (2025).
Version
Final published version
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s00454-024-00708-z
