Advertisement

On the Total Number of Bends for Planar Octilinear Drawings

  • Michael A. BekosEmail author
  • Michael Kaufmann
  • Robert Krug
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9644)

Abstract

An octilinear drawing of a planar graph is one in which each edge is drawn as a sequence of horizontal, vertical and diagonal at \(45^\circ \) line-segments. For such drawings to be readable, special care is needed in order to keep the number of bends small. As the problem of finding planar octilinear drawings of minimum number of bends is NP-hard, in this paper we focus on upper and lower bounds. From a recent result of Keszegh et al. on the slope number of planar graphs, we can derive an upper bound of \(4n-10\) bends for 8-planar graphs with n vertices. We considerably improve this general bound and corresponding previous ones for triconnected 4-, 5- and 6-planar graphs. We also derive non-trivial lower bounds for these three classes of graphs by a technique inspired by the network flow formulation of Tamassia.

Keywords

Planar Graph Horizontal Segment Blue Edge Horizontal Line Segment Green Edge 
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.

References

  1. 1.
    Badent, M., Brandes, U., Cornelsen, S.: More canonical ordering. J. Graph Algorithms Appl. 15(1), 97–126 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bekos, M.A., Gronemann, M., Kaufmann, M., Krug, R.: Planar octilinear drawings with one bend per edge. J. Graph Algorithms Appl. 19(2), 657–680 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bekos, M.A., Kaufmann, M., Krug, R.: On the total number of bends for planar octilinear drawings. Arxiv report arxiv.org/abs/1512.04866 (2014)
  4. 4.
    Biedl, T.C.: New lower bounds for orthogonal graph drawings. In: Brandenburg, F.J. (ed.) GD 1995. LNCS, vol. 1027, pp. 28–39. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  5. 5.
    Biedl, T.C., Kant, G.: A better heuristic for orthogonal graph drawings. Comput. Geom. 9(3), 159–180 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bodlaender, H.L., Tel, G.: A note on rectilinearity and angular resolution. J. Graph Algorithms Appl. 8(1), 89–94 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    De Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Di Giacomo, E., Liotta, G., Montecchiani, F.: The Planar Slope Number of Subcubic Graphs. In: Pardo, A., Viola, A. (eds.) LATIN 2014. LNCS, vol. 8392, pp. 132–143. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  9. 9.
    Felsner, S.: Schnyder woods or how to draw a planar graph? In: Geometric Graphs and Arrangements, pp. 17–42. Advanced Lectures in Mathematics, Vieweg/Teubner Verlag (2004)Google Scholar
  10. 10.
    Fößmeier, U., Heß, C., Kaufmann, M.: On improving orthogonal drawings: the 4M-algorithm. In: Whitesides, S.H. (ed.) GD 1998. LNCS, vol. 1547, pp. 125–137. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  11. 11.
    Garg, A., Tamassia, R.: On the computational complexity of upward and rectilinear planarity testing. SIAM J. Comput. 31(2), 601–625 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kant, G.: Drawing planar graphs using the lmc-ordering. In: FOCS, pp. 101–110. IEEE (1992)Google Scholar
  13. 13.
    Kant, G.: Hexagonal grid drawings. In: Mayr, E.W. (ed.) WG 1992. LNCS, vol. 657, pp. 263–276. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  14. 14.
    Keszegh, B., Pach, J., Pálvölgyi, D.: Drawing planar graphs of bounded degree with few slopes. SIAM J. Discrete Math. 27(2), 1171–1183 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Liu, Y., Morgana, A., Simeone, B.: A linear algorithm for 2-bend embeddings of planar graphs in the two-dimensional grid. Discrete Appl. Math. 81(1–3), 69–91 (1998)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Nöllenburg, M.: Automated drawings of metro maps. Technical Report 2005–25, Fakultät für Informatik, Universität Karlsruhe (2005)Google Scholar
  17. 17.
    Tamassia, R.: On embedding a graph in the grid with the minimum number of bends. SIAM J. Comput. 16(3), 421–444 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Tamassia, R., Tollis, I.G., Vitter, J.S.: Lower bounds for planar orthogonal drawings of graphs. Inf. Process. Lett. 39(1), 35–40 (1991)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Michael A. Bekos
    • 1
    Email author
  • Michael Kaufmann
    • 1
  • Robert Krug
    • 1
  1. 1.Wilhelm-Schickard-Institut für InformatikUniversität TübingenTübingenGermany

Personalised recommendations