Approximate proximity drawings

  • Authors:

  • William Evans;Emden Gansner;Michael Kaufmann;Giuseppe Liotta;Henk Meijer;Andreas Spillner

  • Affiliations:

  • University of British Columbia, Canada;AT&T Research Labs, NJ, United States;Universität Tübingen, Germany;Universití degli Studi di Perugia, Italy;Roosevelt Academy, The Netherlands;Universität Greifswald, Germany

  • Venue:
  • Computational Geometry: Theory and Applications
  • Year:
  • 2013

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Abstract

We introduce and study a generalization of the well-known region of influence proximity drawings, called (@e"1,@e"2)-proximity drawings. Intuitively, given a definition of proximity and two real numbers @e"1=0 and @e"2=0, an (@e"1,@e"2)-proximity drawing of a graph is a planar straight-line drawing @C such that: (i) for every pair of adjacent vertices u, v, their proximity region ''shrunk'' by the multiplicative factor 11+@e"1 does not contain any vertices of @C; (ii) for every pair of non-adjacent vertices u, v, their proximity region ''expanded'' by the factor (1+@e"2) contains some vertices of @C other than u and v. In particular, the locations of the vertices in such a drawing do not always completely determine which edges must be present/absent, giving us some freedom of choice. We show that this generalization significantly enlarges the family of representable planar graphs for relevant definitions of proximity drawings, including Gabriel drawings, Delaunay drawings, and @b-drawings, even for arbitrarily small values of @e"1 and @e"2. We also study the extremal case of (0,@e"2)-proximity drawings, which generalize the well-known weak proximity drawing paradigm.