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Balanced Circle Packings for Planar Graphs

  • Md. Jawaherul Alam
  • David Eppstein
  • Michael T. Goodrich
  • Stephen G. Kobourov
  • Sergey Pupyrev
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8871)

Abstract

We study balanced circle packings and circle-contact representations for planar graphs, where the ratio of the largest circle’s diameter to the smallest circle’s diameter is polynomial in the number of circles. We provide a number of positive and negative results for the existence of such balanced configurations.

References

  1. 1.
    Alam, J., Eppstein, D., Goodrich, M.T., Kobourov, S.G., Pupyrev, S.: Balanced circle packings for planar graphs. Arxiv report arxiv.org/abs/1408.4902 (2014)Google Scholar
  2. 2.
    Bannister, M.J., Devanny, W.E., Eppstein, D., Goodrich, M.T.: The Galois complexity of graph drawing: Why numerical solutions are ubiquitous for force-directed, spectral, and circle packing drawings. In: Duncan, C., Symvonis, A. (eds.) GD 2014. LNCS, vol. 8871, pp. 149–161. Springer, Heidelberg (2014)Google Scholar
  3. 3.
    Bern, M., Eppstein, D.: Optimal Möbius transformations for information visualization and meshing. In: Dehne, F., Sack, J.-R., Tamassia, R. (eds.) WADS 2001. LNCS, vol. 2125, pp. 14–25. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  4. 4.
    Breu, H., Kirkpatrick, D.G.: Unit disk graph recognition is NP-hard. Comput. Geom. Th. Appl. 9(1-2), 3–24 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Brightwell, G., Scheinerman, E.: Representations of planar graphs. SIAM J. Discrete Math. 6(2), 214–229 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Chen, G., Yu, X.: Long cycles in 3-connected graphs. J. Comb. Theory B 86(1), 80–99 (2002)CrossRefzbMATHGoogle Scholar
  7. 7.
    Collins, C.R., Stephenson, K.: A circle packing algorithm. Comput. Geom. Th. Appl. 25(3), 233–256 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Dolev, D., Leighton, T., Trickey, H.: Planar embedding of planar graphs. Advances in Computing Research 2, 147–161 (1984)Google Scholar
  9. 9.
    Duncan, C.A., Gansner, E.R., Hu, Y.F., Kaufmann, M., Kobourov, S.G.: Optimal polygonal representation of planar graphs. Algorithmica 63(3), 672–691 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Eppstein, D.: Planar Lombardi drawings for subcubic graphs. In: Didimo, W., Patrignani, M. (eds.) GD 2012. LNCS, vol. 7704, pp. 126–137. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  11. 11.
    Eppstein, D., Holten, D., Löffler, M., Nöllenburg, M., Speckmann, B., Verbeek, K.: Strict confluent drawing. In: Wismath, S., Wolff, A. (eds.) GD 2013. LNCS, vol. 8242, pp. 352–363. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  12. 12.
    de Fraysseix, H., de Mendez, P.O., Rosenstiehl, P.: On triangle contact graphs. Combinatorics, Probability & Computing 3(2), 233–246 (1994)CrossRefzbMATHGoogle Scholar
  13. 13.
    Gilbert, E.N., Moore, E.F.: Variable-length binary encodings. Bell System Technical Journal 38(4), 933–967 (1959)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Gonçalves, D., Lévêque, B., Pinlou, A.: Triangle contact representations and duality. Discrete Comput. Geom. 48(1), 239–254 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Hliněný, P.: Classes and recognition of curve contact graphs. J. Comb. Theory B 74(1), 87–103 (1998)CrossRefzbMATHGoogle Scholar
  16. 16.
    Koebe, P.: Kontaktprobleme der konformen Abbildung. Ber. Sächs. Akad. Wiss. Leipzig, Math.-Phys. Kl. 88, 141–164 (1936)Google Scholar
  17. 17.
    Luo, F.: Rigidity of polyhedral surfaces, III. Geometry & Topology 15(4), 2299–2319 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Malitz, S.M., Papakostas, A.: On the angular resolution of planar graphs. SIAM J. Discrete Math. 7(2), 172–183 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Mohar, B.: A polynomial time circle packing algorithm. Discrete Math. 117(1-3), 257–263 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Nešetřil, J., Ossona de Mendez, P.: Sparsity: Graphs, Structures, and Algorithms. Springer (2012)Google Scholar
  21. 21.
    Nievergelt, J., Reingold, E.M.: Binary search trees of bounded balance. SIAM J. Comput. 2, 33–43 (1973)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Md. Jawaherul Alam
    • 1
  • David Eppstein
    • 2
  • Michael T. Goodrich
    • 2
  • Stephen G. Kobourov
    • 1
  • Sergey Pupyrev
    • 1
    • 3
  1. 1.Department of Computer ScienceUniversity of ArizonaTucsonUSA
  2. 2.Department of Computer ScienceUniversity of CaliforniaIrvineUSA
  3. 3.Institute of Mathematics and Computer ScienceUral Federal UniversityRussia

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