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Approximate Proximity Drawings

  • William Evans
  • Emden R. Gansner
  • Michael Kaufmann
  • Giuseppe Liotta
  • Henk Meijer
  • Andreas Spillner
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7034)

Abstract

We introduce and study a generalization of the well-known region of influence proximity drawings, called 1,ε 2)-proximity drawings. Intuitively, given a definition of proximity and two real numbers ε 1 ≥ 0 and ε 2 ≥ 0, an (ε 1,ε 2)-proximity drawing of a graph is a planar straight-line drawing Γ such that: (i) for every pair of adjacent vertices u,v, their proximity region “shrunk” by the multiplicative factor \(\frac{1}{1+\varepsilon_1}\) does not contain any vertices of Γ; (ii) for every pair of non-adjacent vertices u,v, their proximity region “blown-up” by the factor (1 + ε 2) contains some vertices of Γ other than u and v. We show that by using this generalization, we can significantly enlarge the family of the representable planar graphs for relevant definitions of proximity drawings, including Gabriel drawings, Delaunay drawings, and β-drawings, even for arbitrarily small values of ε 1 and ε 2. We also study the extremal case of (0,ε 2)-proximity drawings, which generalizes the well-known weak proximity drawing model.

Keywords

Planar Graph Delaunay Triangulation Adjacent Vertex Outerplanar Graph Planar Embedding 
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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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • William Evans
    • 1
  • Emden R. Gansner
    • 2
  • Michael Kaufmann
    • 3
  • Giuseppe Liotta
    • 4
  • Henk Meijer
    • 5
  • Andreas Spillner
    • 6
  1. 1.University of British ColumbiaCanada
  2. 2.AT&T Research LabsUS
  3. 3.Universität TübingenGermany
  4. 4.Università degli Studi di PerugiaItaly
  5. 5.Roosevelt AcademyThe Netherlands
  6. 6.Universität GreifswaldGermany

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