Monotone Simultaneous Embeddings of Paths in d Dimensions

  • David Bremner
  • Olivier Devillers
  • Marc Glisse
  • Sylvain Lazard
  • Giuseppe Liotta
  • Tamara Mchedlidze
  • Sue Whitesides
  • Stephen Wismath
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9801)


We study the following problem: Given k paths that share the same vertex set, is there a simultaneous geometric embedding of these paths such that each individual drawing is monotone in some direction? We prove that, for any dimension d, there is a set of \(d+1\) paths that does not admit a monotone simultaneous geometric embedding.


  1. 1.
    Aichholzer, O., Hackl, T., Lutteropp, S., Mchedlidze, T., Pilz, A., Vogtenhuber, B.: Monotone simultaneous embeddings of upward planar digraphs. J. Graph Algorithms Appl. 19(1), 87–110 (2015)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Angelini, P., Colasante, E., Battista, G.D., Frati, F., Patrignani, M.: Monotone drawings of graphs. J. Graph Algorithms Appl. 16(1), 5–35 (2012)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Angelini, P., Didimo, W., Kobourov, S.G., Mchedlidze, T., Roselli, V., Symvonis, A., Wismath, S.K.: Monotone drawings of graphs with fixed embedding. Algorithmica 71(2), 233–257 (2015)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Angelini, P., Geyer, M., Kaufmann, M., Neuwirth, D.: On a tree and a path with no geometric simultaneous embedding. J. Graph Algorithms Appl. 16(1), 37–83 (2012)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Asinowski, A.: Suballowable sequences and geometric permutations. Discret. Math. 308(20), 4745–4762 (2008)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Blaäsius, T., Kobourov, S.G., Rutter, I.: Simultaneous embedding of planar graphs. In: Tamassia, R. (ed.) Handbook on Graph Drawing and Visualization. Chapman and Hall/CRC, Boca Raton (2013)Google Scholar
  7. 7.
    Braß, P., Cenek, E., Duncan, C.A., Efrat, A., Erten, C., Ismailescu, D.P., Kobourov, S.G., Lubiw, A., Mitchell, J.S.B.: On simultaneous planar graph embeddings. Comput. Geom. 36(2), 117–130 (2007)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bremner, D., Devillers, O., Glisse, M., Lazard, S., Liotta, G., Mchedlidze, T., Whitesides, S., Wismath, S.: Monotone simultaneous embeddings of paths in \(\mathbb{R}^d\) (2016).
  9. 9.
    Erten, C., Kobourov, S.G.: Simultaneous embedding of planar graphs with few bends. J. Graph Algorithms Appl. 9(3), 347–364 (2005)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Felsner, S., Igamberdiev, A., Kindermann, P., Klemz, B., Mchedlidze, T., Scheucher, M.: Strongly monotone drawings of planar graphs. In: Fekete, S., Lubiw, A. (eds.) 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), vol. 51, pp. 37:1–37:15. Dagstuhl, Germany (2016). Schloss Dagstuhl-Leibniz-Zentrum fuer InformatikGoogle Scholar
  11. 11.
    Giacomo, E.D., Didimo, W., Liotta, G., Meijer, H., Wismath, S.K.: Planar and quasi-planar simultaneous geometric embedding. Comput. J. 58(11), 3126–3140 (2015)CrossRefMATHGoogle Scholar
  12. 12.
    Hossain, M.I., Rahman, M.S.: Straight-line monotone grid drawings of series-parallel graphs. Discret. Math. Algorithms Appl. 7(2), 1550007-1–1550007-12 (2015)MathSciNetMATHGoogle Scholar
  13. 13.
    Kindermann, P., Schulz, A., Spoerhase, J., Wolff, A.: On monotone drawings of trees. In: Duncan, C., Symvonis, A. (eds.) GD 2014. LNCS, vol. 8871, pp. 488–500. Springer, Heidelberg (2014). doi:10.1007/978-3-662-45803-7_41 Google Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • David Bremner
    • 1
  • Olivier Devillers
    • 2
  • Marc Glisse
    • 3
  • Sylvain Lazard
    • 2
  • Giuseppe Liotta
    • 4
  • Tamara Mchedlidze
    • 5
  • Sue Whitesides
    • 6
  • Stephen Wismath
    • 7
  1. 1.University of New BrunswickFrederictonCanada
  2. 2.Inria, CNRS, University of LorraineNancyFrance
  3. 3.InriaSaclayFrance
  4. 4.University of PerugiaPerugiaItaly
  5. 5.Karlsruhe Institute of TechnologyKarlsruheGermany
  6. 6.University of VictoriaVictoriaCanada
  7. 7.University of LethbridgeLethbridgeCanada

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