Document Open Access Logo

Rectilinear Crossing Number of Graphs Excluding a Single-Crossing Graph as a Minor

Authors Vida Dujmović , Camille La Rose

PDF
Thumbnail PDF

File

LIPIcs.GD.2024.37.pdf
  • Filesize: 0.78 MB
  • 18 pages

Document Identifiers

Author Details

Vida Dujmović
  • University of Ottawa, Canada
Camille La Rose
  • University of Ottawa, Canada

Funding

  • Dujmović, Vida: Funded by NSERC and a University of Ottawa Research Chair.
  • La Rose, Camille: Funded by NSERC.

Cite AsGet BibTex

Vida Dujmović and Camille La Rose. Rectilinear Crossing Number of Graphs Excluding a Single-Crossing Graph as a Minor. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 37:1-37:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.GD.2024.37

Abstract

The rectilinear crossing number of G is the minimum number of crossings in a straight-line drawing of G. A single-crossing graph is a graph whose crossing number is at most one. We prove that every n-vertex graph G that excludes a single-crossing graph as a minor has rectilinear crossing number O(Δ n), where Δ is the maximum degree of G. This dependence on n and Δ is best possible. The result applies, for example, to K₅-minor-free graphs, and bounded treewidth graphs. Prior to our work, the only bounded degree minor-closed families known to have linear rectilinear crossing number were bounded degree graphs of bounded treewidth as well as bounded degree K_{3,3}-minor-free graphs. In the case of bounded treewidth graphs, our O(Δ n) result is again tight and it improves on the previous best known bound of O(Δ² n) by Wood and Telle, 2007.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Mathematics of computing → Graphs and surfaces
  • Theory of computation → Computational geometry
Keywords
  • (rectilinear) crossing number
  • graph minors
  • maximum degree
  • clique-sums

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Géza . Crossing number of toroidal graphs. In Topics in discrete mathematics, pages 581-590. Springer, 2006. Google Scholar
  2. Martin Aigner and Günter M. Ziegler. Proofs from The Book. Springer, third edition, 2004. Google Scholar
  3. M. Ajtai, V. Chvátal, M.M. Newborn, and E. Szemerédi. Crossing-free subgraphs. In Theory and Practice of Combinatorics, volume 60 of North-Holland Mathematics Studies, pages 9-12. North-Holland, 1982. Google Scholar
  4. Sandeep N. Bhatt and F. Thomson Leighton. A framework for solving VLSI graph layout problems. J. Comput. System Sci., 28(2):300-343, 1984. URL: https://doi.org/10.1016/0022-0000(84)90071-0.
  5. Daniel Bienstock and Nathaniel Dean. Bounds for rectilinear crossing numbers. Journal of graph theory, 17(3):333-348, 1993. URL: https://doi.org/10.1002/JGT.3190170308.
  6. Hans L. Bodlaender. A partial k-arboretum of graphs with bounded treewidth. Theoretical Computer Science, 209(1):1-45, 1998. URL: https://doi.org/10.1016/S0304-3975(97)00228-4.
  7. Sergio Cabello. Hardness of approximation for crossing number. Discret. Comput. Geom., 49(2):348-358, 2013. URL: https://doi.org/10.1007/S00454-012-9440-6.
  8. 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: https://doi.org/10.1137/120872310.
  9. Erin W. Chambers and David Eppstein. Flows in one-crossing-minor-free graphs. J. Graph Algorithms Appl., 17(3):201-220, 2013. URL: https://doi.org/10.7155/JGAA.00291.
  10. Julia Chuzhoy and Zihan Tan. A subpolynomial approximation algorithm for graph crossing number in low-degree graphs. In STOC '22: 54th Annual ACM SIGACT Symposium on Theory of Computing, pages 303-316. ACM, 2022. URL: https://doi.org/10.1145/3519935.3519984.
  11. Erik D. Demaine, Fedor V. Fomin, Mohammad Taghi Hajiaghayi, and Dimitrios M. Thilikos. Subexponential parameterized algorithms on bounded-genus graphs and H-minor-free graphs. J. ACM, 52(6):866-893, 2005. URL: https://doi.org/10.1145/1101821.1101823.
  12. Erik D. Demaine, Mohammad Taghi Hajiaghayi, and Dimitrios M. Thilikos. Exponential speedup of fixed-parameter algorithms for classes of graphs excluding single-crossing graphs as minors. Algorithmica, 41(4):245-267, 2005. URL: https://doi.org/10.1007/S00453-004-1125-Y.
  13. Josep Dıaz, Norman Do, Maria J Serna, and Nicholas C Wormald. Bounds on the max and min bisection of random cubic and random 4-regular graphs. Theoretical computer science, 307(3):531-547, 2003. URL: https://doi.org/10.1016/S0304-3975(03)00236-6.
  14. Hristo N Djidjev and Imrich Vrt’o. Crossing numbers and cutwidths. Journal of Graph Algorithms and Applications. v7, pages 245-251, 2006. Google Scholar
  15. Hristo N Djidjev and Imrich Vrt’o. Planar crossing numbers of graphs of bounded genus. Discrete & Computational Geometry, 48(2):393-415, 2012. URL: https://doi.org/10.1007/S00454-012-9430-8.
  16. Vida Dujmović, Ken-ichi Kawarabayashi, Bojan Mohar, and David R. Wood. Tight upper bounds on the crossing number in a minor-closed class. CoRR, abs/1807.11617, 2018. URL: https://arxiv.org/abs/1807.11617.
  17. David Eppstein and Vijay V. Vazirani. NC algorithms for computing a perfect matching and a maximum flow in one-crossing-minor-free graphs. SIAM J. Comput., 50(3):1014-1033, 2021. URL: https://doi.org/10.1137/19M1256221.
  18. P. Erdos and R. K. Guy. Crossing number problems. The American Mathematical Monthly, 80(1):52-58, 1973. Google Scholar
  19. M. R. Garey and D. S. Johnson. Crossing number is np-complete. SIAM Journal on Algebraic Discrete Methods, 4(3):312-316, 1983. Google Scholar
  20. Rudolf Halin. S-functions for graphs. J. Geometry, 8(1-2):171-186, 1976. Google Scholar
  21. Fáry István. On straight-line representation of planar graphs. Acta scientiarum mathematicarum, 11(229-233):2, 1948. Google Scholar
  22. Károly J., János Pach, and Géza Tóth. Planar crossing numbers of graphs embeddable in another surface. Int. J. Found. Comput. Sci., 17(5):1005-1016, 2006. URL: https://doi.org/10.1142/S0129054106004236.
  23. Ken-ichi Kawarabayashi and Buce Reed. Computing crossing number in linear time. In Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing, STOC '07, pages 382-390. ACM, 2007. URL: https://doi.org/10.1145/1250790.1250848.
  24. Alexandr V. Kostochka. Lower bound of the Hadwiger number of graphs by their average degree. Combinatorica, 4(4):307-316, 1984. URL: https://doi.org/10.1007/BF02579141.
  25. AV Kostochka and LS Mel'nikov. On bounds of the bisection width of cubic graphs. In Annals of Discrete Mathematics, volume 51, pages 151-154. Elsevier, 1992. Google Scholar
  26. F. Thomson Leighton. Complexity Issues in VLSI. MIT Press, 1983. Google Scholar
  27. F. Thomson Leighton. New lower bound techniques for VLSI. Math. Systems Theory, 17(1):47-70, 1984. URL: https://doi.org/10.1007/BF01744433.
  28. Frank Thomson Leighton. Complexity issues in VLSI: optimal layouts for the shuffle-exchange graph and other networks. MIT Press, Cambridge, MA, USA, 1983. Google Scholar
  29. Bernard Montaron. An improvement of the crossing number bound. J. Graph Theory, 50(1):43-54, 2005. URL: https://doi.org/10.1002/JGT.20090.
  30. Nagi H. Nahas. On the crossing number of K_m,n. Electron. J. Comb., 10, 2003. URL: https://doi.org/10.37236/1748.
  31. János Pach, Radoš Radoičić, Gábor Tardos, and Géza Tóth. Improving the crossing lemma by finding more crossings in sparse graphs. Discrete Comput. Geom., 36(4):527-552, 2006. URL: https://doi.org/10.1007/S00454-006-1264-9.
  32. R Bruce Richter and J Širáň. The crossing number of K_3, n in a surface. Journal of Graph Theory, 21(1):51-54, 1996. URL: https://doi.org/10.1002/(SICI)1097-0118(199601)21:1%3C51::AID-JGT7%3E3.0.CO;2-L.
  33. Neil Robertson and Paul Seymour. Excluding a graph with one crossing. In Graph structure theory. Proceedings of the AMS-IMS-SIAM joint summer research conference on graph minors, pages 669-675. Providence, RI: American Mathematical Society, 1993. Google Scholar
  34. Neil Robertson and Paul D. Seymour. Graph minors. II. algorithmic aspects of tree-width. J. Algorithms, 7(3):309-322, 1986. URL: https://doi.org/10.1016/0196-6774(86)90023-4.
  35. Neil Robertson and P.D Seymour. Graph minors. V. excluding a planar graph. Journal of Combinatorial Theory, Series B, 41(1):92-114, 1986. URL: https://doi.org/10.1016/0095-8956(86)90030-4.
  36. Neil Robertson and P.D Seymour. Graph minors. XVI. excluding a non-planar graph. Journal of Combinatorial Theory. Series B, 89(1):43-76, 2003. URL: https://doi.org/10.1016/S0095-8956(03)00042-X.
  37. Marcus Schaefer. The graph crossing number and its variants: a survey. Electron. J. Comb., DS21:90, 2013. Google Scholar
  38. Farhad Shahrokhi, Ondrej Sýkora, László A. Székely, and Imrich Vrto. Bounds for convex crossing numbers. In Tandy J. Warnow and Binhai Zhu, editors, Computing and Combinatorics, 9th Annual International Conference, COCOON 2003, volume 2697 of LNCS, pages 487-495. Springer, 2003. URL: https://doi.org/10.1007/3-540-45071-8_49.
  39. Andrew Thomason. An extremal function for contractions of graphs. Math. Proc. Cambridge Philos. Soc., 95(2):261-265, 1984. URL: https://doi.org/10.1017/S0305004100061521.
  40. Klaus Wagner. Bemerkungen zum vierfarbenproblem. Jahresbericht der Deutschen Mathematiker-Vereinigung, 46:26-32, 1936. Google Scholar
  41. David R. Wood and Jan Arne. Planar decompositions and the crossing number of graphs with an excluded minor. New York J. Math., 13:117-146, 2007. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact