Line Crossing Minimization on Metro Maps

  • Michael A. Bekos
  • Michael Kaufmann
  • Katerina Potika
  • Antonios Symvonis
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4875)


We consider the problem of drawing a set of simple paths along the edges of an embedded underlying graph G = (V,E), so that the total number of crossings among pairs of paths is minimized. This problem arises when drawing metro maps, where the embedding of G depicts the structure of the underlying network, the nodes of G correspond to train stations, an edge connecting two nodes implies that there exists a railway line which connects them, whereas the paths illustrate the lines connecting terminal stations. We call this the metro-line crossing minimization problem (MLCM).

In contrast to the problem of drawing the underlying graph nicely, MLCM has received fewer attention. It was recently introduced by Benkert et. al in [4] . In this paper, as a first step towards solving MLCM in arbitrary graphs, we study path and tree networks. We examine several variations of the problem for which we develop algorithms for obtaining optimal solutions.


Metro Maps Crossing Minimization Lines Paths Trees 


  1. 1.
    Asquith, M., Gudmundsson, J., Merrick, D.: An ILP for the line ordering problem. Technical Report PA006288, National ICT Australia (2007)Google Scholar
  2. 2.
    Battista, G.D., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing: Algorithms for the Visualization of Graphs. Prentice-Hall, Englewood Cliffs (1999)MATHGoogle Scholar
  3. 3.
    Bekos, M.A., Kaufmann, M., Potika, K., Symvonis, A.: Line crossing minimization on metro maps. Technical Report WSI-2007-03, University of Tübingen (2007)Google Scholar
  4. 4.
    Benkert, M., Nöllenburg, M., Uno, T., Wolff, A.: Minimizing intra-edge crossings in wiring diagrams and public transport maps. In: Kaufmann, M., Wagner, D. (eds.) GD 2006. LNCS, vol. 4372, pp. 270–281. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  5. 5.
    Hong, S.-H., Merrick, D., Nascimento, H.A.D.d.: The metro map layout problem. In: Churcher, N., Churcher, C. (eds.) 2004. Australasian Symposium on Information Visualisation, CRPIT, ACS, vol. 35, pp. 91–100 (2004)Google Scholar
  6. 6.
    Kaufmann, M., Wagner, D. (eds.): Drawing Graphs. LNCS, vol. 2025. Springer, Heidelberg (2001)MATHGoogle Scholar
  7. 7.
    Masuda, S., Nakajima, K., Kashiwabara, T., Fujisawa, T.: Crossing minimization in linear embeddings of graphs. IEEE Trans. Comput. 39(1), 124–127 (1990)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Nöllenburg, M., Wolff, A.: A mixed-integer program for drawing high-quality metro maps. In: Healy, P., Nikolov, N.S. (eds.) GD 2005. LNCS, vol. 3843, pp. 321–333. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  9. 9.
    Stott, J.M., Rodgers, P.: Metro map layout using multicriteria optimization. In: Proc. 8th International Conference on Information Visualisation, pp. 355–362. IEEE Computer Society, Los Alamitos (2004)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Michael A. Bekos
    • 1
  • Michael Kaufmann
    • 2
  • Katerina Potika
    • 1
  • Antonios Symvonis
    • 1
  1. 1.School of Applied Mathematics & Physical SciencesNational Technical University of AthensZografouGreece
  2. 2.Institute for InformaticsUniversity of TübingenTübingenGermany

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