Straight-Line Drawings on Restricted Integer Grids in Two and Three Dimensions

(Extended Abstract)
  • Stefan Felsner
  • Giuseppe Liotta
  • Stephen Wismath
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2265)


This paper investigates the following question: Given an integer grid ϕ, where ϕ is a proper subset of the integer plane or a proper subset of the integer 3d space, which graphs admit straight-line crossingfree drawings with vertices located at the grid points of ϕ? We characterize the trees that can be drawn on a two dimensional c · n × к grid, where к and c are given integer constants, and on a two dimensional grid consisting of к parallel horizontal lines of infinite length. Motivated by the results on the plane we investigate restrictions of the integer grid in 3 dimensions and show that every outerplanar graph with n vertices can be drawn crossing-free with straight lines in linear volume on a grid called a prism. This prism consists of 3n integer grid points and is universal — it supports all outerplanar graphs of n vertices. This is the first algorithm that computes crossing-free straight line 3d drawings in linear volume for a non-trivial family of planar graphs. We also show that there exist planar graphs that cannot be drawn on the prism and that the extension to a n × 2 × 2 integer grid, called a box, does not admit the entire class of planar graphs.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Stefan Felsner
    • 1
  • Giuseppe Liotta
    • 2
  • Stephen Wismath
    • 3
  1. 1.Fachbereich Mathematik und InformatikFreie Universität BerlinBerlinGermany
  2. 2.Dipartimento di Ingegneria Elettronica e dell’InformazioneUniversità degli Studi di PerugiaVia DurantiItaly
  3. 3.Dept. of Mathematics and Computer ScienceU. of LethbridgeAlbertaCanada

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