On Hardness of the Joint Crossing Number

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9472)

Abstract

The Joint Crossing Number problem asks for a simultaneous embedding of two disjoint graphs into one surface such that the number of edge crossings (between the two graphs) is minimized. It was introduced by Negami in 2001 in connection with diagonal flips in triangulations of surfaces, and subsequently investigated in a general form for small-genus surfaces. We prove that all of the commonly considered variants of this problem are NP-hard already in the orientable surface of genus 6, by a reduction from a special variant of the anchored crossing number problem of Cabello and Mohar.

References

  1. 1.
    Archdeacon, D., Bonnington, C.P.: Two maps on one surface. J. Graph Theory 36(4), 198–216 (2001)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Cabello, S., Mohar, B.: Crossing number and weighted crossing number of near-planar graphs. Algorithmica 60(3), 484–504 (2011)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Cabello, S., Mohar, B.: Adding one edge to planar graphs makes crossing number and 1-planarity hard. SIAM J. Comput. 42(5), 1803–1829 (2013)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Hliněný, P., Salazar, G.: On hardness of the joint crossing number (2015). Full version arXiv:1509.01787
  5. 5.
    Negami, S.: Diagonal flips in triangulations on closed surfaces, estimating upper bounds. Yokohama Math. J. 45(2), 113–124 (1998)MathSciNetMATHGoogle Scholar
  6. 6.
    Negami, S.: Crossing numbers of graph embedding pairs on closed surfaces. J. Graph Theory 36(1), 8–23 (2001)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bruce Richter, R., Salazar, G.: Two maps with large representativity on one surface. J. Graph Theory 50(3), 234–245 (2005)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Schaefer, M.: The graph crossing number and its variants: a survey. Electronic Journal of Combinatorics, #DS21, May 15, 2014Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Faculty of InformaticsMasaryk University BrnoBrnoCzech Republic
  2. 2.Instituto de FisicaUniversidad Autonoma de San Luis PotosiSan Luis PotosiMexico

Personalised recommendations