Low Ply Drawings of Trees

  • Patrizio Angelini
  • Michael A. BekosEmail author
  • Till Bruckdorfer
  • Jaroslav HančlJr.
  • Michael Kaufmann
  • Stephen Kobourov
  • Antonios Symvonis
  • Pavel Valtr
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9801)


We consider the recently introduced model of low ply graph drawing, in which the ply-disks of the vertices do not have many common overlaps, which results in a good distribution of the vertices in the plane. The ply-disk of a vertex in a straight-line drawing is the disk centered at it whose radius is half the length of its longest incident edge. The largest number of ply-disks having a common overlap is called the ply-number of the drawing.

We focus on trees. We first consider drawings of trees with constant ply-number, proving that they may require exponential area, even for stars, and that they may not even exist for bounded-degree trees. Then, we turn our attention to drawings with logarithmic ply-number and show that trees with maximum degree 6 always admit such drawings in polynomial area.


  1. 1.
    Alam, M.J., Eppstein, D., Goodrich, M.T., Kobourov, S.G., Pupyrev, S.: Balanced circle packings for planar graphs. In: Duncan, C.A., Symvonis, A. (eds.) GD 2014. LNCS, vol. 8871, pp. 125–136. Springer, Heidelberg (2014). doi: 10.1007/978-3-662-45803-7_11 Google Scholar
  2. 2.
    Angelini, P., Bekos, M.A., Bruckdorfer Jr., T.J.H., Kaufmann, M., Kobourov, S., Symvonis, A., Valtr, P.: Low ply drawings of trees. CoRR abs/1608.08538v2 (2016)Google Scholar
  3. 3.
    Breu, H., Kirkpatrick, D.G.: Unit disk graph recognition is NP-hard. Comput. Geometry 9(1–2), 3–24 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Buchheim, C., Chimani, M., Gutwenger, C., Jünger, M., Mutzel, P.: Crossings and planarization. In: Tamassia, R. (ed.) Handbook on Graph Drawing and Visualization, pp. 43–85. Chapman and Hall/CRC, Boca Raton (2013)Google Scholar
  5. 5.
    Di Giacomo, E., Didimo, W., Hong, S., Kaufmann, M., Kobourov, S.G., Liotta, G., Misue, K., Symvonis, A., Yen, H.: Low ply graph drawing. In: 6th International Conference on Information, Intelligence, Systems and Applications, IISA 2015, pp. 1–6. IEEE (2015)Google Scholar
  6. 6.
    Eades, P., Hong, S.: Symmetric graph drawing. In: Tamassia, R. (ed.) Handbook on Graph Drawing and Visualization, pp. 87–113. Chapman and Hall/CRC, Boca Raton (2013)Google Scholar
  7. 7.
    Eppstein, D., Goodrich, M.T.: Studying (non-planar) road networks through an algorithmic lens. In: GIS 2008, pp. 1–10. ACM (2008)Google Scholar
  8. 8.
    Fekete, S.P., Houle, M.E., Whitesides, S.: The wobbly logic engine: Proving hardness of non-rigid geometric graph representation problems. In: DiBattista, G. (ed.) GD 1997. LNCS, vol. 1353, pp. 272–283. Springer, Heidelberg (1997). doi: 10.1007/3-540-63938-1_69 CrossRefGoogle Scholar
  9. 9.
    Hliněný, P.: Contact graphs of curves. In: Brandenburg, F. (ed.) Graph Drawing. LNCS, vol. 1027, pp. 312–323. Springer, Heidelberg (1995). doi: 10.1007/BFb0021814 CrossRefGoogle Scholar
  10. 10.
    Hliněný, P.: Classes and recognition of curve contact graphs. J. Combin. Theory Ser. B 74(1), 87–103 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Koebe, P.: Kontaktprobleme der konformen Abbildung. Berichte über die Verhandlungen der Sächsischen Akad. der Wissenschaften zu Leipzig. Math.-Phys. Klasse 88, 141–164 (1936)Google Scholar
  12. 12.
    Liotta, G.: Proximity drawings. In: Tamassia, R. (ed.) Handbook on Graph Drawing and Visualization, pp. 115–154. Chapman and Hall/CRC, Boca Raton (2013)Google Scholar
  13. 13.
    Sleator, D.D., Tarjan, R.E.: A data structure for dynamic trees. J. Comput. Syst. Sci. 26(3), 362–391 (1983)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Patrizio Angelini
    • 1
  • Michael A. Bekos
    • 1
    Email author
  • Till Bruckdorfer
    • 1
  • Jaroslav HančlJr.
    • 2
  • Michael Kaufmann
    • 1
  • Stephen Kobourov
    • 3
  • Antonios Symvonis
    • 4
  • Pavel Valtr
    • 2
  1. 1.Institut für InformatikUniversität TübingenTübingenGermany
  2. 2.Department of Applied MathematicsCharles University (KAM)PragueCzech Republic
  3. 3.Department for Computer ScienceUniversity of ArizonaTucsonUSA
  4. 4.School of Applied Mathematical and Physical SciencesNTUAAthensGreece

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