Drawing Graphs in the Plane with a Prescribed Outer Face and Polynomial Area

  • Erin W. Chambers
  • David Eppstein
  • Michael T. Goodrich
  • Maarten Löffler
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6502)


We study the classic graph drawing problem of drawing a planar graph using straight-line edges with a prescribed convex polygon as the outer face. Unlike previous algorithms for this problem, which may produce drawings with exponential area, our method produces drawings with polynomial area. In addition, we allow for collinear points on the boundary, provided such vertices do not create overlapping edges. Thus, we solve an open problem of Duncan et al., which, when combined with their work, implies that we can produce a planar straight-line drawing of a combinatorially-embedded genus-g graph with the graph’s canonical polygonal schema drawn as a convex polygonal external face.


Hull Stein Rote 


  1. 1.
    Bárány, I., Rote, G.: Strictly convex drawings of planar graphs. Documenta Mathematica 11, 369–391, (2006) arXiv:cs/0507030 , http://www.math.uiuc.edu/documenta/vol-11/13.html MathSciNetMATHGoogle Scholar
  2. 2.
    Becker, B., Hotz, G.: On the optimal layout of planar graphs with fixed boundary. SIAM J. Comput. 16(5), 946–972 (1987), doi:10.1137/0216061MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Chrobak, M., Goodrich, M.T., Tamassia, R.: Convex drawings of graphs in two and three dimensions. In: Proc. 12th ACM Symp. Comput. Geom., pp. 319–328 (1996), doi:10.1145/237218.237401Google Scholar
  4. 4.
    Chrobak, M., Kant, G.: Convex grid drawings of 3-connected planar graphs. Internat. J. Comput. Geom. Appl. 7(3), 211–223 (1997), doi:10.1142/S0218195997000144MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chrobak, M., Payne, T.H.: A linear-time algorithm for drawing a planar graph on a grid. Inf. Proc. Lett. 54(4), 241–246 (1995), doi:10.1016/0020-0190(95)00020-DMathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Davidson, R., Harel, D.: Drawing graphs nicely using simulated annealing. ACM Trans. Graph. 15(4), 301–331 (1996), doi:10.1145/234535.234538CrossRefGoogle Scholar
  7. 7.
    Dhandapani, R.: Greedy drawings of triangulations. Discrete Comput. Geom. 43(2), 375–392 (2010), doi:10.1007/s00454-009-9235-6MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing. Prentice Hall, Upper Saddle River (1999)MATHGoogle Scholar
  9. 9.
    Duncan, C.A., Goodrich, M.T., Kobourov, S.G.: Planar drawings of higher-genus graphs. In: Eppstein, D., Gansner, E.R. (eds.) GD 2009. LNCS, vol. 5849, Springer, Heidelberg (2010), doi:10.1007/978-3-642-11805-0_7Google Scholar
  10. 10.
    Fáry, I.: On straight-line representation of planar graphs. Acta Sci. Math. (Szeged) 11, 229–233 (1948)MathSciNetMATHGoogle Scholar
  11. 11.
    de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990), doi:10.1007/BF02122694MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Fruchterman, T.M.J., Reingold, E.M.: Graph drawing by force-directed placement. Softw. Pract. Exp. 21(11), 1129–1164 (1991), doi:10.1002/spe.4380211102CrossRefGoogle Scholar
  13. 13.
    Gajer, P., Goodrich, M.T., Kobourov, S.G.: A multi-dimensional approach to force-directed layouts of large graphs. Comput. Geom. Theory Appl. 29(1), 3–18 (2004), doi:10.1016/j.comgeo.2004.03.014MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Kant, G.: Drawing planar graphs using the canonical ordering. Algorithmica 16(1), 4–32 (1996), doi:10.1007/BF02086606MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Lazarus, F., Pocchiola, M., Vegter, G., Verroust, A.: Computing a canonical polygonal schema of an Orientable Triangulated Surface. In: Proc. 17th ACM Symp. Comput. Geom., pp. 80–89 (2001), doi:10.1145/378583.378630Google Scholar
  16. 16.
    Schnyder, W.: Embedding planar graphs on the grid. In: Proc. 1st ACM-SIAM Symp. Discrete Algorithms, pp. 138–148 (1990), http://portal.acm.org/citation.cfm?id=320191
  17. 17.
    Stein, S.K.: Convex maps. Proc. Amer. Math. Soc. 2(3), 464–466 (1951), doi:10.1090/S0002-9939-1951-0041425-5MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Sugiyama, K., Misue, K.: Graph drawing by the magnetic spring model. J. Visual Lang. Comput. 6(3), 217–231 (1995), doi:10.1006/jvlc.1995.1013CrossRefGoogle Scholar
  19. 19.
    Tutte, W.T.: Convex representations of graphs. Proc. London Math. Soc. 10(38), 304–320 (1960), doi:10.1112/plms/s3-10.1.304Google Scholar
  20. 20.
    Tutte, W.T.: How to draw a graph. Proc. London Math. Soc. 13(52), 743–768 (1963), doi:10.1112/plms/s3-13.1.743Google Scholar
  21. 21.
    Wagner, K.: Bemerkungen zum Vierfarbenproblem. Jber. Deutsch. Math.-Verein. 46, 26–32 (1936)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Erin W. Chambers
    • 1
  • David Eppstein
    • 2
  • Michael T. Goodrich
    • 2
  • Maarten Löffler
    • 2
  1. 1.Dept. of Math and Computer ScienceSaint Louis Univ.USA
  2. 2.Computer Science Dept.University of CaliforniaIrvineUSA

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