Genus, Treewidth, and Local Crossing Number

  • Vida Dujmović
  • David EppsteinEmail author
  • David R. Wood
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9411)


We consider relations between the size, treewidth, and local crossing number (maximum number of crossings per edge) of graphs embedded on topological surfaces. We show that an n-vertex graph embedded on a surface of genus g with at most k crossings per edge has treewidth \(O(\sqrt{(g+1)(k+1)n})\) and layered treewidth \(O((g+1)k)\), and that these bounds are tight up to a constant factor. As a special case, the k-planar graphs with n vertices have treewidth \(O(\sqrt{(k+1)n})\) and layered treewidth \(O(k+1)\), which are tight bounds that improve a previously known \(O((k+1)^{3/4}n^{1/2})\) treewidth bound. Additionally, we show that for \(g<m\), every m-edge graph can be embedded on a surface of genus g with \(O((m/(g+1))\log ^2 g)\) crossings per edge, which is tight to a polylogarithmic factor.


Planar Graph Tree Decomposition Grid Graph Graph Minor Cyclomatic Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This research was initiated at the Workshop on Graphs and Geometry held at the Bellairs Research Institute in 2015. Vida Dujmović was supported by NSERC. David Eppstein was supported in part by NSF grant CCF-1228639. David Wood was supported by the Australian Research Council.


  1. 1.
    Dvorák, Z., Norin, S.: Treewidth of graphs with balanced separations. Electronic preprint arXiv:  1408.3869 (2014)
  2. 2.
    Reed, B.A.: Tree width and tangles: a new connectivity measure and some applications. In: Bailey, R.A. (ed.) Surveys in Combinatorics. London Mathematical Society Lecture Note Series, vol. 241, pp. 87–162. Cambridge University Press, Cambridge (1997)Google Scholar
  3. 3.
    Schaefer, M.: The graph crossing number and its variants: a survey. Electron. J. Combin. DS21 (2014)Google Scholar
  4. 4.
    Grigoriev, A., Bodlaender, H.L.: Algorithms for graphs embeddable with few crossings per edge. Algorithmica 49(1), 1–11 (2007)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Guy, R.K., Jenkyns, T., Schaer, J.: The toroidal crossing number of the complete graph. J. Comb. Theor. 4, 376–390 (1968)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Gilbert, J.R., Hutchinson, J.P., Tarjan, R.E.: A separator theorem for graphs of bounded genus. J. Algorithms 5(3), 391–407 (1984)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Dujmović, V., Morin, P., Wood, D.R.: Layered separators in minor-closed families with applications. Electronic preprint arXiv:  1306.1595 (2013)
  8. 8.
    Shahrokhi, F., Székely, L.A., Sýkora, O., Vrt’o, I.: Drawings of graphs on surfaces with few crossings. Algorithmica 16(1), 118–131 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Halin, R.: \(S\)-functions for graphs. J. Geometry 8(1–2), 171–186 (1976)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Robertson, N., Seymour, P.D.: Graph minors. II. algorithmic aspects of tree-width. J. Algorithms 7(3), 309–322 (1986)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Eppstein, D.: Diameter and treewidth in minor-closed graph families. Algorithmica 27, 275–291 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Hoory, S., Linial, N., Wigderson, A.: Expander graphs and their applications. Bull. Am. Math. Soc. 43(4), 439–561 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Grohe, M., Marx, D.: On tree width, bramble size, and expansion. J. Combin. Theory Ser. B 99(1), 218–228 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Leighton, T., Rao, S.: Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms. J. ACM 46(6), 787–832 (1999)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Vida Dujmović
    • 1
  • David Eppstein
    • 2
    Email author
  • David R. Wood
    • 3
  1. 1.School of Computer Science and Electrical EngineeringUniversity of OttawaOttawaCanada
  2. 2.Department of Computer ScienceUniversity of CaliforniaIrvineUSA
  3. 3.School of Mathematical SciencesMonash UniversityMelbourneAustralia

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