Level Planar Embedding in Linear Time

  • Michael Jünger
  • Sebastian Leipert
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1731)

Abstract

In a level directed acyclic graph G = (V;E) the vertex set V is partitioned into k ≤ |V | levels V 1; V 2... V k such that for each edge (u, v) ∈ E with uV i and v ∈; V j we have i < j. The level planarity testing problem is to decide if G can be drawn in the plane such that for each level V i, all vV i are drawn on the line l i = {(x, k - i) | x ∈ ℝ}, the edges are drawn monotonically with respect to the vertical direction, and no edges intersect except at their end vertices. In order to draw a level planar graph without edge crossings, a level planar embedding of the level graph has to be computed. Level planar embeddings are characterized by linear orderings of the vertices in each V i (1 ≤ ik). We present an O(|V |) time algorithm for embedding level planar graphs. This approach is based on a level planarity test by Jünger, Leipert, and Mutzel [6].

References

  1. 1.
    K. Booth and G. Lueker. Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms. Journal of Computer and System Sciences, 13:335–379, 1976.MATHMathSciNetGoogle Scholar
  2. 2.
    N. Chiba, T. Nishizeki, S. Abe, and T. Ozawa. A linear algorithm for embedding planar graphs using PQ-trees. Journal of Computer and System Sciences, 30:54–76, 1985.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    G. Di Battista and R. Tamassia. Algorithms for plane representations of acyclic digraphs. Theoretical Computer Science, 61:175–198, 1988.CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    G. Di Battista, R. Tamassia, and I. G. Tollis. Constrained visibility representations of graphs. Information Processing Letters, 41:1–7, 1992.MATHCrossRefGoogle Scholar
  5. 5.
    L. S. Heath and S. V. Pemmaraju. Recognizing leveled-planar dags in linear time. In F. J. Brandenburg, editor, Proc. Graph Drawing’ 95, volume 1027 of Lecture Notes in Computer Science, pages 300–311. Springer Verlag, 1995.CrossRefGoogle Scholar
  6. 6.
    M. Jünger, S. Leipert, and P. Mutzel. Level planarity testing in linear time. In S. Whitesides, editor, Graph Drawing’ 98, volume 1547 of Lecture Notes in Computer Science, pages 224–237. Springer Verlag, 1998.CrossRefGoogle Scholar
  7. 7.
    S. Leipert. Level Planarity Testing and Embedding in Linear Time. PhD thesis, Universität zu Köln, 1998.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Michael Jünger
    • 1
  • Sebastian Leipert
    • 1
  1. 1.Universität zu KölnInstitut für InformatikKölnGermany

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