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Simultaneous Embedding with Two Bends per Edge in Polynomial Area

  • Frank Kammer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4059)

Abstract

The simultaneous embedding problem is, given two planar graphs G 1=(V,E 1) and G 2=(V,E 2), to find planar embeddings ϕ(G 1) and ϕ(G 2) such that each vertex vV is mapped to the same point in ϕ(G 1) and in ϕ(G 2). This article presents a linear-time algorithm for the simultaneous embedding problem such that edges are drawn as polygonal chains with at most two bends and all vertices and all bends of the edges are placed on a grid of polynomial size. An extension of this problem with so-called fixed edges is also considered.

A further linear-time algorithm of this article solves the following problem: Given a planar graph G and a set of distinct points, find a planar embedding for G that maps each vertex to one of the given points. The solution presented also uses at most two bends per edge and a grid whose size is polynomial in the size of the grid that includes all given points. An example shows two bends per edge to be optimal.

Keywords

Planar Graph Hamilton Path Dual Graph Outer Face Outerplanar Graph 
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.

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References

  1. 1.
    Bern, M., Gilbert, J.R.: Drawing the planar dual. Information Processing Letters 43(1), 7–13 (1992)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Brass, P., Cenek, E., Duncan, C., Efrat, A., Erten, C., Ismailescu, D., Kobourov, S., Lubiw, A., Mitchell, J.: On Simultaneous Planar Graph Embeddings. In: Dehne, F., Sack, J.-R., Smid, M. (eds.) WADS 2003. LNCS, vol. 2748, pp. 219–230. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  3. 3.
    Chiba, N., Nishizeki, T.: Arboricity and subgraph listing algorithms. SIAM J. Comput. 14, 210–223 (1985)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Chiba, N., Nishizeki, T.: The hamiltonian cycle problem is linear-time solvable for 4-connected planar graphs. Journal of Algorithm 10(2), 187–211 (1989)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Dillencourt, M.B., Eppstein, D., Hirschberg, D.S.: Geometric thickness of complete graphs. Journal of Graph Algorithms and Applications 4(3), 5–17 (2000)MATHMathSciNetGoogle Scholar
  6. 6.
    Erten, C., Kobourov, S.G.: Simultaneous embedding of planar graphs with few bends. In: Pach, J. (ed.) GD 2004. LNCS, vol. 3383, pp. 195–205. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  7. 7.
    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)MATHMathSciNetGoogle Scholar
  8. 8.
    Mutzel, P., Odental, T., Scharbrodt, M.: The thickness of graphs: a survey. Graphs Combin. 14(1), 59–73 (1998)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Pach, J., Wenger, R.: Embedding planar graphs at fixed vertex locations. Graphs and Combinatorics 17, 717–728 (2001)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Frank Kammer
    • 1
  1. 1.Institut für InformatikUniversität AugsburgAugsburgGermany

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