Advertisement

Drawing Planar Graphs with Few Geometric Primitives

  • Gregor Hültenschmidt
  • Philipp Kindermann
  • Wouter Meulemans
  • André Schulz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10520)

Abstract

We define the visual complexity of a plane graph drawing to be the number of basic geometric objects needed to represent all its edges. In particular, one object may represent multiple edges (e.g., one needs only one line segment to draw two collinear edges of the same vertex). Let n denote the number of vertices of a graph. We show that trees can be drawn with 3n / 4 straight-line segments on a polynomial grid, and with n / 2 straight-line segments on a quasi-polynomial grid. Further, we present an algorithm for drawing planar 3-trees with \((8n\,-\,17)/3\) segments on an \(O(n)\,\times \,O(n^2)\) grid. This algorithm can also be used with a small modification to draw maximal outerplanar graphs with 3n / 2 edges on an \(O(n)\,\times \,O(n^2)\) grid. We also study the problem of drawing maximal planar graphs with circular arcs and provide an algorithm to draw such graphs using only \((5n\,-\,11)/3\) arcs. This provides a significant improvement over the lower bound of 2n for line segments for a nontrivial graph class.

References

  1. 1.
    Bonichon, N., Le Saëc, B., Mosbah, M.: Wagner’s theorem on realizers. In: Widmayer, P., Eidenbenz, S., Triguero, F., Morales, R., Conejo, R., Hennessy, M. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 1043–1053. Springer, Heidelberg (2002).  https://doi.org/10.1007/3-540-45465-9_89 CrossRefGoogle Scholar
  2. 2.
    Brehm, E.: 3-orientations and Schnyder 3-tree-decompositions. In: Master’s Thesis, Freie Universität Berlin (2000). http://page.math.tu-berlin.de/~felsner/Diplomarbeiten/brehm.ps.gz
  3. 3.
    Chaplick, S., Fleszar, K., Lipp, F., Ravsky, A., Verbitsky, O., Wolff, A.: Drawing graphs on few lines and few planes. In: Hu, Y., Nöllenburg, M. (eds.) GD 2016. LNCS, vol. 9801, pp. 166–180. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-50106-2_14 CrossRefGoogle Scholar
  4. 4.
    Chaplick, S., Fleszar, K., Lipp, F., Ravsky, A., Verbitsky, O., Wolff, A.: The complexity of drawing graphs on few lines and few planes. In: Ellen, F., Kolokolova, A., Sack, J.R. (eds.) Algorithms and Data Structures. LNCS, vol. 10389, pp. 265–276. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-62127-2_23 CrossRefGoogle Scholar
  5. 5.
    de Fraysseix, H., de Mendez, P.O.: On topological aspects of orientations. Discrete Math. 229(1–3), 57–72 (2001).  https://doi.org/10.1016/S0012-365X(00)00201-6 CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    de Fraysseix, H., Pach, J., Pollack, R.: Small sets supporting Fary embeddings of planar graphs. In: Simon, J. (ed.) Proceedings of 20th Annual ACM Symposium on Theory of Computing (STOC 1988), pp. 426–433. ACM, 1988.  https://doi.org/10.1145/62212.62254
  7. 7.
    de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990).  https://doi.org/10.1007/BF02122694 CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Dujmović, V., Eppstein, D., Suderman, M., Wood, D.R.: Drawings of planar graphs with few slopes and segments. Comput. Geom. Theory Appl. 38(3), 194–212 (2007).  https://doi.org/10.1016/j.comgeo.2006.09.002 CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Durocher, S., Mondal, D.: Drawing plane triangulations with few segments. In: He, M., Zeh, N. (eds.) Proceedings of 26th Canadian Conference on Computational Geometry (CCCG 2014), Carleton University, pp. 40–45 (2014). http://www.cccg.ca/proceedings/2014/papers/paper06.pdf
  10. 10.
    Durocher, S., Mondal, D., Nishat, R.I., Whitesides, S.: A note on minimum-segment drawings of planar graphs. J. Graph Algorithms Appl. 17(3), 301–328 (2013).  https://doi.org/10.7155/jgaa.00295 CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Felsner, S., Trotter, W.T.: Posets and planar graphs. J. Graph Theory 49(4), 273–284 (2005).  https://doi.org/10.1002/jgt.20081 CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Hültenschmidt, G., Kindermann, P., Meulemans, W., Schulz, A.: Drawing planar graphs with few geometric primitives (2017). Arxiv report 1703.01691. arXiv:1703.01691
  13. 13.
    Igamberdiev, A., Meulemans, W., Schulz, A.: Drawing planar cubic 3-connected graphs with few segments: algorithms and experiments. In: Di Giacomo, E., Lubiw, A. (eds.) GD 2015. LNCS, vol. 9411, pp. 113–124. Springer, Cham (2015).  https://doi.org/10.1007/978-3-319-27261-0_10 CrossRefGoogle Scholar
  14. 14.
    Lick, D.R., White, A.T.: \(k\)-degenerate graphs. Can. J. Math. 22, 1082–1096 (1970).  https://doi.org/10.4153/CJM-1970-125-1 CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Mondal, D.: Visualizing graphs: optimization and trade-offs. Ph.D. thesis, University of Manitoba (2016). http://hdl.handle.net/1993/31673
  16. 16.
    Mondal, D., Nishat, R.I., Biswas, S., Rahman, M.S.: Minimum-segment convex drawings of 3-connected cubic plane graphs. J. Comb. Optim. 25(3), 460–480 (2013).  https://doi.org/10.1007/s10878-011-9390-6 CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Schnyder, W.: Embedding planar graphs on the grid. In: Johnson, D.S. (ed.) Proceedings of 1st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 1990), pp 138–148. SIAM (1990). http://dl.acm.org/citation.cfm?id=320191
  18. 18.
    Schulz, A.: Drawing graphs with few arcs. J. Graph Algorithms Appl. 19(1), 393–412 (2015).  https://doi.org/10.7155/jgaa.00366 CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Tarjan, R.E.: Linking and cutting trees. In: Data Structures and Network Algorithms, pp. 59–70. SIAM (1983).  https://doi.org/10.1137/1.9781611970265.ch5
  20. 20.
    Wade, G.A., Chu, J.: Drawability of complete graphs using a minimal slope set. Comput. J. 37(2), 139–142 (1994).  https://doi.org/10.1093/comjnl/37.2.139 CrossRefGoogle Scholar
  21. 21.
    Zhang, H., He, X.: Canonical ordering trees and their applications in graph drawing. Discrete Comput. Geom. 33(2), 321–344 (2005).  https://doi.org/10.1007/s00454-004-1154-y CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Gregor Hültenschmidt
    • 1
  • Philipp Kindermann
    • 1
  • Wouter Meulemans
    • 2
  • André Schulz
    • 1
  1. 1.LG Theoretische InformatikFernUniversität in HagenHagenGermany
  2. 2.Department of Mathematics and Computer ScienceTU EindhovenEindhovenThe Netherlands

Personalised recommendations