Crossing Number is Hard for Kernelization

Authors Petr Hlinený, Marek Dernár

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Petr Hlinený
Marek Dernár

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Petr Hlinený and Marek Dernár. Crossing Number is Hard for Kernelization. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 42:1-42:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


The graph crossing number problem, cr(G)<=k, asks for a drawing of a graph G in the plane with at most k edge crossings. Although this problem is in general notoriously difficult, it is fixed-parameter tractable for the parameter k [Grohe, STOC 2001]. This suggests a closely related question of whether this problem has a polynomial kernel, meaning whether every instance of cr(G)<=k can be in polynomial time reduced to an equivalent instance of size polynomial in k (and independent of |G|). We answer this question in the negative. Along the proof we show that the tile crossing number problem of twisted planar tiles is NP-hard, which has been an open problem for some time, too, and then employ the complexity technique of cross-composition. Our result holds already for the special case of graphs obtained from planar graphs by adding one edge.
  • crossing number; tile crossing number; parameterized complexity; polynomial kernel; cross-composition


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  1. Michael J. Bannister, David Eppstein, and Joseph A. Simons. Fixed parameter tractability of crossing minimization of almost-trees. In Graph Drawing - GD 2013, volume 8242 of Lecture Notes in Computer Science, pages 340-351. Springer, 2013. URL:
  2. Hans L. Bodlaender, Bart M. P. Jansen, and Stefan Kratsch. Kernelization lower bounds by cross-composition. SIAM J. Discrete Math., 28(1):277-305, 2014. URL:
  3. Drago Bokal, Mojca Bracic, Marek Derňár, and Petr Hliněný. On degree properties of crossing-critical families of graphs. In Graph Drawing and Network Visualization - GD 2015, volume 9411 of Lecture Notes in Computer Science, pages 75-86. Springer, 2015. Google Scholar
  4. Drago Bokal, Bogdan Oporowski, R. Bruce Richter, and Gelasio Salazar. Characterizing 2-crossing-critical graphs. Manuscript, 171 pages., 2013. Google Scholar
  5. Sergio Cabello. Hardness of approximation for crossing number. Discrete & Computational Geometry, 49(2):348-358, 2013. URL:
  6. Sergio Cabello and Bojan Mohar. Adding one edge to planar graphs makes crossing number and 1-planarity hard. SIAM J. Comput., 42(5):1803-1829, 2013. URL:
  7. Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013. URL:
  8. J. Flum and M. Grohe. Parameterized Complexity Theory. Springer, 2006. Google Scholar
  9. Lance Fortnow and Rahul Santhanam. Infeasibility of instance compression and succinct PCPs for NP. J. Comput. Syst. Sci., 77(1):91-106, 2011. URL:
  10. M. R. Garey and D. S. Johnson. Crossing number is NP-complete. SIAM J. Alg. Discr. Meth., 4:312-316, 1983. Google Scholar
  11. Martin Grohe. Computing crossing numbers in quadratic time. In Proceedings on 33rd Annual ACM Symposium on Theory of Computing, July 6-8, 2001, Heraklion, Crete, Greece, pages 231-236. ACM, 2001. URL:
  12. Petr Hliněný. Crossing number is hard for cubic graphs. J. Comb. Theory, Ser. B, 96(4):455-471, 2006. URL:
  13. Ken-ichi Kawarabayashi and Bruce A. Reed. Computing crossing number in linear time. In Proceedings of the 39th Annual ACM Symposium on Theory of Computing, San Diego, California, USA, June 11-13, 2007, pages 382-390. ACM, 2007. URL:
  14. Martin Kochol. Construction of crossing-critical graphs. Discrete Mathematics, 66(3):311-313, 1987. URL:
  15. B. Mohar and C. Thomassen. Graphs on Surfaces. Johns Hopkins University Press, Baltimore, 2001. Google Scholar
  16. Michael J. Pelsmajer, Marcus Schaefer, and Daniel Stefankovic. Crossing numbers of graphs with rotation systems. Algorithmica, 60(3):679-702, 2011. URL:
  17. Benny Pinontoan and R. Bruce Richter. Crossing numbers of sequences of graphs II: planar tiles. Journal of Graph Theory, 42(4):332-341, 2003. URL:
  18. Benny Pinontoan and R. Bruce Richter. Crossing numbers of sequences of graphs I: General tiles. The Australasian Journal of Combinatorics, 30:197-206, 2004. Google Scholar
  19. R. Bruce Richter and Carsten Thomassen. Minimal graphs with crossing number at least k. J. Comb. Theory, Ser. B, 58(2):217-224, 1993. URL:
  20. Marcus Schaefer. The graph crossing number and its variants: A survey. Electronic Journal of Combinatorics, #DS21, May 15, 2014. Google Scholar