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Simultaneous Embedding of Planar Graphs with Few Bends

  • Cesim Erten
  • Stephen G. Kobourov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3383)

Abstract

We present an O(n) time algorithm for simultaneous embedding of pairs of planar graphs on the O(n 2O(n 2) grid, with at most three bends per edge, where n is the number of vertices. For the case when the input graphs are both trees, only one bend per edge is required. We also describe an O(n) time algorithm for simultaneous embedding with fixed-edges for tree-path pairs on the O(nO(n 2) grid with at most one bend per tree-edge and no bends along path edges.

Keywords

Planar Graph Hamiltonian Cycle Input Graph Planar Embedding Geometric Thickness 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Brandes, U., Corman, S.R.: Visual unrolling of network evolution and the analysis of dynamic discourse. In: IEEE Symposium on Information Visualization (INFOVIS 2002), pp. 145–151 (2002)Google Scholar
  2. 2.
    Brass, P., Cenek, E., Duncan, C.A., Efrat, A., Erten, C., Ismailescu, D., Kobourov, S.G., Lubiw, A., Mitchell, J.S.B.: On simultaneous graph embedding. In: 8th Workshop on Algorithms and Data Structures, pp. 243–255 (2003)Google Scholar
  3. 3.
    Chiba, N., Nishizeki, T.: Arboricity and subgraph listing algorithms. SIAM J. Comput. 14, 210–223 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Chiba, N., Nishizeki, T.: The hamiltonian cycle problem is linear-time solvable for 4-connected planar graphs. Journal of Algorithms 10(2), 187–211 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Dillencourt, M.B., Eppstein, D., Hirschberg, D.S.: Geometric thickness of complete graphs. Journal of Graph Algorithms and Applications 4(3), 5–17 (2000)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Duncan, C.A., Eppstein, D., Kobourov, S.G.: The geometric thickness of low degree graphs. In: 20th Annual ACM-SIAM Symposium on Computational Geometry (SCG), pp. 340–346 (2004)Google Scholar
  8. 8.
    Erten, C., Kobourov, S.G., Navabia, A., Le., V.: Simultaneous graph drawing: Layout algorithms and visualization schemes. In: 11th Symposium on Graph Drawing (GD), pp. 437–449 (2003)Google Scholar
  9. 9.
    Fáry, I.: On straight lines representation of planar graphs. Acta Scientiarum Mathematicarum 11, 229–233 (1948)Google Scholar
  10. 10.
    Hopcroft, J., Tarjan, R.E.: Efficient planarity testing. Journal of the ACM 21(4), 549–568 (1974)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Kaufmann, M., Wiese, R.: Embedding vertices at points: Few bends suffice for planar graphs. Journal of Graph Algorithms and Applications 6(1), 115–129 (2002)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Mutzel, P., Odenthal, T., Scharbrodt, M.: The thickness of graphs: a survey. Graphs Combin. 14(1), 59–73 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Pach, J., Wenger, R.: Embedding planar graphs at fixed vertex locations. Graphs and Combinatorics 17, 717–728 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Schnyder, W.: Embedding planar graphs on the grid. In: Proceedings of the 1st ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 138–148 (1990)Google Scholar
  15. 15.
    Stein, S.K.: Convex maps. Proceedings of the American Mathematical Society 2(3), 464–466 (1951)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Tutte, W.T.: How to draw a graph. Proc. London Math. Society 13(52), 743–768 (1963)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Wagner, K.: Bemerkungen zum vierfarbenproblem. Jahresbericht der Deutschen Mathematiker-Vereinigung 46, 26–32 (1936)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Cesim Erten
    • 1
  • Stephen G. Kobourov
    • 1
  1. 1.Department of Computer ScienceUniversity of Arizona 

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