International Symposium on Graph Drawing and Network Visualization

Graph Drawing and Network Visualization pp 423-429 | Cite as

Linear-Size Universal Point Sets for One-Bend Drawings

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9411)

Abstract

For every integer \(n\ge 4\), we construct a planar point set \(S_n\) of size \(6n-10\) such that every n-vertex planar graph G admits a plane embedding in which the vertices are mapped to points in \(S_n\), and every edge is either a line segment or a polyline with one bend, where the bend point is also in \(S_n\).

References

  1. 1.
    Bannister, M.J., Cheng, Z., Devanny, W.E., Eppstein, D.: Superpatterns and universal point sets. J. Graph Algorithms Appl. 18(2), 177–209 (2014)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Cardinal, J., Hoffmann, M., Kusters, V., Tóth, C.D., Wettstein, M.: Arc diagrams, flip distances, and Hamiltonian triangulations. In: Mayr, E.W., Ollinger, N. (eds.) Proceedings of 32nd STACS. LiPIcs, vol. 30, pp. 197–210. Leibniz-Zentrum für Informatik, Dagstuhl (2015)Google Scholar
  3. 3.
    Chrobak, M., Kant, G.: Convex grid drawings of 3-connected planar graphs. Internat. J. Comput. Geom. Appl. 7, 211–223 (1997)MathSciNetCrossRefGoogle Scholar
  4. 4.
    de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Di Battista, G., Frati, F.: Small area drawings of outerplanar graphs. Algorithmica 54(1), 25–53 (2009)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Dolev, D., Leighton, F.T., Trickey, H.: Planar embedding of planar graphs. In: Preparata, F. (ed.) Advances in Computing Research, vol. 2, pp. 147–161. JAI Press Inc., London (1984)Google Scholar
  7. 7.
    Dujmović, V., Evans, W., Lazard, S., Lenhart, W., Liotta, G., Rappaport, D., Wismath, S.: On point-sets that support planar graphs. Comput. Geom. Theory Appl. 46(1), 29–50 (2013)CrossRefMATHGoogle Scholar
  8. 8.
    Everett, H., Lazard, S., Liotta, G., Wismath, S.: Universal sets of \(n\) points for one-bend drawings of planar graphs with \(n\) vertices. Discrete Comput. Geom. 43(2), 272–288 (2010)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Fáry, I.: On straight lines representation of plane graphs. Acta Scientiarum Mathematicarum (Szeged) 11, 229–233 (1948)MATHGoogle Scholar
  10. 10.
    Frati, F.: Lower bounds on the area requirements of series-parallel graphs. Discrete Math. Theoret. Comput. Sci. 12(5), 139–174 (2010)MathSciNetMATHGoogle Scholar
  11. 11.
    Frati, F., Patrignani, M.: A note on minimum-area straight-line drawings of planar graphs. In: Hong, S.-H., Nishizeki, T., Quan, W. (eds.) GD 2007. LNCS, vol. 4875, pp. 339–344. Springer, Heidelberg (2008) CrossRefGoogle Scholar
  12. 12.
    Di Giacomo, E., Didimo, W., Liotta, G., Wismath, S.K.: Curve-constrained drawings of planar graphs. Comput. Geom. Theory Appl. 30(1), 1–23 (2005)CrossRefMATHGoogle Scholar
  13. 13.
    Di Giacomo, E., Didimo, W., Liotta, G.: Spine and radial drawings. In: Tamassia, R. (ed.) Handbook of Graph Drawing and Visualization, Chap. 8, pp. 247–284. CRC Press, Boca Raton (2013)Google Scholar
  14. 14.
    Kurowski, M.: A 1.235 lower bound on the number of points needed to draw all \(n\)-vertex planar graphs. Inf. Process. Lett. 92, 95–98 (2004)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Schnyder, W.: Embedding planar graphs in the grid. In: Proceedings of the 1st Symposium on Discrete Algorithms, pp. 138–147. ACM Press, New York, NY (1990)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Information and Computing SciencesUtrecht UniversityUtrechtThe Netherlands
  2. 2.Department of MathematicsCalifornia State University NorthridgeLos AngelesUSA

Personalised recommendations