Unit Contact Representations of Grid Subgraphs with Regular Polytopes in 2D and 3D

  • Linda Kleist
  • Benjamin Rahman
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8871)

Abstract

We present a strategy to construct unit proper contact representations (UPCR) for subgraphs of certain highly symmetric grids. This strategy can be applied to obtain graphs admitting UPCRs with squares and cubes, whose recognition is NP-complete.

We show that subgraphs of the square grid allow for UPCR with squares which strengthens the previously known cube representation. Indeed, we give UPCR for subgraphs of a d-dimensional grid with d-cubes. Additionally, we show that subgraphs of the triangular grid admit a UPCR with cubes, implying that the same holds for each subgraph of an Archimedean grid. Considering further polygons, we construct UPCR with regular 3k-gons of the hexagonal grid and UPCR with regular 4k-gons of the square grid.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alam, M.J., Kobourov, S.G., Pupyrev, S., Toeniskoetter, J.: Happy edges: Threshold-coloring of regular lattices. In: Ferro, A., Luccio, F., Widmayer, P. (eds.) FUN 2014. LNCS, vol. 8496, pp. 28–39. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  2. 2.
    Alam, M.J., Chaplick, S., Fijavž, G., Kaufmann, M., Kobourov, S.G., Pupyrev, S.: Threshold-coloring and unit-cube contact representation of graphs. In: Brandstädt, A., Jansen, K., Reischuk, R. (eds.) WG 2013. LNCS, vol. 8165, pp. 26–37. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  3. 3.
    Bremner, D., Evans, W., Frati, F., Heyer, L., Kobourov, S.G., Lenhart, W.J., Liotta, G., Rappaport, D., Whitesides, S.H.: On representing graphs by touching cuboids. In: Didimo, W., Patrignani, M. (eds.) GD 2012. LNCS, vol. 7704, pp. 187–198. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  4. 4.
    Breu, H., Kirkpatrick, D.G.: On the complexity of recognizing intersection and touching graphs of disks. In: Brandenburg, F.J. (ed.) GD 1995. LNCS, vol. 1027, pp. 88–98. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  5. 5.
    Czyzowicz, J., Kranakis, E., Krizanc, D., Urrutia, J.: Discrete realizations of contact and intersection graphs. International Journal of Pure and Applied Mathematics 13(4), 429 (2004)MATHMathSciNetGoogle Scholar
  6. 6.
    Felsner, S., Francis, M.C.: Contact representations of planar graphs with cubes. In: Proceedings of the 27th Annual ACM Symposium on Computational Geometry, pp. 315–320. ACM (2011)Google Scholar
  7. 7.
    Gansner, E.R., Hu, Y.F., Kaufmann, M., Kobourov, S.G.: Optimal polygonal representation of planar graphs. In: López-Ortiz, A. (ed.) LATIN 2010. LNCS, vol. 6034, pp. 417–432. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  8. 8.
    Gonçalves, D., Lévêque, B., Pinlou, A.: Triangle contact representations and duality. Discrete & Computational Geometry 48(1), 239–254 (2012)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Hliněnỳ, P., Kratochvíl, J.: Representing graphs by disks and balls (a survey of recognition-complexity results). Discrete Mathematics 229(1-3), 101–124 (2001)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Koebe, P.: Kontaktprobleme der konformen abbildung. Berichte über die Verhandlungen der Sächsischen Akademien der Wissenschaften zu Leipzig, Math.-Phys. Kl. 88, 141–164 (1936)Google Scholar
  11. 11.
    Schramm, O.: Combinatorically prescribed packings and applications to conformal and quasiconformal maps. Ph. D. thesis. Princeton University (1990)Google Scholar
  12. 12.
    Schramm, O.: Square tilings with prescribed combinatorics. Israel Journal of Mathematics 84(1-2), 97–118 (1993)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Zhao, L.: The kissing number of the regular polygon. Discrete Mathematics 188(1), 293–296 (1998)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Zhao, L., Xu, J.: The kissing number of the regular pentagon. Discrete Mathematics 252(1), 293–298 (2002)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Linda Kleist
    • 1
  • Benjamin Rahman
    • 1
  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany

Personalised recommendations