Abstract
LetG be a graph ofn vertices that can be drawn in the plane by straight-line segments so that nok+1 of them are pairwise crossing. We show thatG has at mostc k nlog2k−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 mappingf∶R 2→R 2 withf(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.
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Communicated by G. Di Battista and R. Tamassia.
The first author was supported by NSF Grant CCR-91-22103, PSC-CUNY Research Award 663472, and OTKA-4269. An extended abstract of this paper was presented at the 10th Annual ACM Symposium on Computational Geometry, Stony Brook, NY, 1994.
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Pach, J., Shahrokhi, F. & Szegedy, M. Applications of the crossing number. Algorithmica 16, 111–117 (1996). https://doi.org/10.1007/BF02086610
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DOI: https://doi.org/10.1007/BF02086610