The Planar Slope Number of Planar Partial 3-Trees of Bounded Degree

  • Vít Jelínek
  • Eva Jelínková
  • Jan Kratochvíl
  • Bernard Lidický
  • Marek Tesař
  • Tomáš Vyskočil
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5849)

Abstract

It is known that every planar graph has a planar embedding where edges are represented by non-crossing straight-line segments. We study the planar slope number, i.e., the minimum number of distinct edge-slopes in such a drawing of a planar graph with maximum degree Δ. We show that the planar slope number of every series-parallel graph of maximum degree three is three. We also show that the planar slope number of every planar partial 3-tree and also every plane partial 3-tree is at most \(2^{{\mathcal O}(\Delta)}\). In particular, we answer the question of Dujmović et al. [Computational Geometry 38 (3), pp. 194–212 (2007)] whether there is a function f such that plane maximal outerplanar graphs can be drawn using at most f(Δ) slopes.

Keywords

graph drawing planar graphs slopes planar slope number 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Vít Jelínek
    • 1
    • 2
  • Eva Jelínková
    • 1
  • Jan Kratochvíl
    • 1
    • 3
  • Bernard Lidický
    • 1
  • Marek Tesař
    • 1
  • Tomáš Vyskočil
    • 1
    • 3
  1. 1.Department of Applied MathematicsCharles University in Prague 
  2. 2.Combinatorics GroupReykjavík University 
  3. 3.Institute for Theoretical Computer ScienceCharles University in Prague 

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