Bundled Crossings in Embedded Graphs

  • Martin FinkEmail author
  • John Hershberger
  • Subhash Suri
  • Kevin Verbeek
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9644)


Edge crossings in a graph drawing are an important factor in the drawing’s quality. However, it is not just the presence of crossings that determines the drawing’s quality: any drawing of a nonplanar graph in the plane necessarily contains crossings, but the geometric structure of those crossings can have a significant impact on the drawing’s readability. In particular, the structure of two disjoint groups of locally parallel edges (bundles) intersecting in a complete crossbar (a bundled crossing) is visually simpler—even if it involves many individual crossings—than an equal number of random crossings scattered in the plane.

In this paper, we investigate the complexity of partitioning the crossings of a given drawing of a graph into a minimum number of bundled crossings. We show that this problem is NP-hard, propose a constant-factor approximation scheme for the case of circular embeddings, where all vertices lie on the outer face, and show that the bundled crossings problem in general graphs is related to a minimum dissection problem.


Outer Face Graph Drawing Metro Line Individual Crossing Outer Cycle 
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.



The research of Martin Fink was partially supported by a fellowship within the Postdoc-Program of the German Academic Exchange Service (DAAD), and by NSF grants CCF-1161495 and CCF-1525817. The research of Subhash Suri was partially supported by NSF grants CCF-1161495 and CCF-1525817.


  1. 1.
    Dahlhaus, E., Johnson, D.S., Papadimitriou, C.H., Seymour, P.D., Yannakakis, M.: The complexity of multiterminal cuts. SIAM J. Comput. 23(4), 864–894 (1994). MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Eppstein, D., van Kreveld, M.J., Mumford, E., Speckmann, B.: Edges and switches, tunnels and bridges. Comput. Geom. 42(8), 790–802 (2009). MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Fink, M., Pupyrev, S., Wolff, A.: Ordering metro lines by block crossings. J. Graph Algorithms Appl. 19(1), 111–153 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Füredi, Z., Palásti, I.: Arrangements of lines with a large number of triangles. Proc. Am. Math. Soc. 92(4), 561–566 (1984). MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Garey, M.R., Johnson, D.S.: Crossing number is NP-complete. SIAM J. Algebraic Discrete Methods 4(3), 312–316 (1983). MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Holten, D.: Hierarchical edge bundles: visualization of adjacency relations in hierarchical data. IEEE Trans. Vis. Comput. Graph. 12(5), 741–748 (2006). CrossRefGoogle Scholar
  7. 7.
    Holten, D., van Wijk, J.J.: Force-directed edge bundling for graph visualization. Comput. Graph. Forum 28(3), 983–990 (2009). CrossRefGoogle Scholar
  8. 8.
    Hu, Y., Shi, L.: A coloring algorithm for disambiguating graph and map drawings. In: Duncan, C., Symvonis, A. (eds.) GD 2014. LNCS, vol. 8871, pp. 89–100. Springer, Heidelberg (2014). Google Scholar
  9. 9.
    Huang, W., Hong, S.H., Eades, P.: Effects of crossing angles. In: Proceedings of 7th International IEEE Asia-Pacific Symposium Information Visualisation (PacificVIS 2008), pp. 41–46 (2008).
  10. 10.
    Schaefer, M.: The graph crossing number and its variants: asurvey. Electron. J. Comb. Dyn. Surv. 21, 1–100 (2013). MathSciNetGoogle Scholar
  11. 11.
    Soltan, V., Gorpinevich, A.: Minimum dissection of a rectilinear polygon with arbitrary holes into rectangles. Discrete Comp. Geom. 9(1), 57–79 (1993). MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Martin Fink
    • 1
    Email author
  • John Hershberger
    • 2
  • Subhash Suri
    • 1
  • Kevin Verbeek
    • 3
  1. 1.Department of Computer ScienceUniversity of CaliforniaSanta BarbaraUSA
  2. 2.Mentor Graphics CorporationWilsonvilleUSA
  3. 3.Department of Mathematics and Computer ScienceTU EindhovenEindhovenThe Netherlands

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