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Drawings of graphs on surfaces with few crossings

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Abstract

We give drawings of a complete graphK n withO(n 4 log2 g/g) many crossings on an orientable or nonorientable surface of genusg ≥ 2. We use these drawings ofK n and give a polynomial-time algorithm for drawing any graph withn vertices andm edges withO(m 2 log2 g/g) many crossings on an orientable or nonorientable surface of genusg ≥ 2. Moreover, we derive lower bounds on the crossing number of any graph on a surface of genusg ≥ 0. The number of crossings in the drawings produced by our algorithm are within a multiplicative factor ofO(log2 g) from the lower bound (and hence from the optimal) for any graph withm ≥ 8n andn 2/m ≤g ≤m/64.

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Authors and Affiliations

  1. Department of Computer Science, University of North Texas, P.O. Box 13886, 76203, Denton, TX, USA

    F. Shahrokhi

  2. Department of Computer Science, Eötvös University, Múzeurn krt. 6-8, 1088, Budapest, Hungary

    L. A. Székely

  3. Institute for Informatics, Slovak Academy of Sciences, Dúbravská 9, P.O. Box 56, 840 00, Bratislava 4, Slovakia

    O. Sýkora & I. Vrt'o

Authors
  1. F. Shahrokhi
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  2. L. A. Székely
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  3. O. Sýkora
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  4. I. Vrt'o
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Additional information

Communicated by G. Di Battista and R. Tamassia.

The research of the third and the fourth authors was partially supported by Grant No. 2/1138/94 of the Slovak Academy of Sciences and by EC Cooperative action IC1000 “Algorithms for Future Technologies” (Project ALTEC). A preliminary version of this paper was presented at WG93 and published in Lecture Notes in Computer Science, Vol. 790, 1993, pp. 388–396.

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Shahrokhi, F., Székely, L.A., Sýkora, O. et al. Drawings of graphs on surfaces with few crossings. Algorithmica 16, 118–131 (1996). https://doi.org/10.1007/BF02086611

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  • DOI: https://doi.org/10.1007/BF02086611

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