Linear-Time Algorithms for Hole-Free Rectilinear Proportional Contact Graph Representations

  • Muhammad Jawaherul Alam
  • Therese Biedl
  • Stefan Felsner
  • Andreas Gerasch
  • Michael Kaufmann
  • Stephen G. Kobourov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7074)

Abstract

In a proportional contact representation of a planar graph, each vertex is represented by a simple polygon with area proportional to a given weight, and edges are represented by adjacencies between the corresponding pairs of polygons. In this paper we study proportional contact representations that use rectilinear polygons without wasted areas (white space). In this setting, the best known algorithm for proportional contact representation of a maximal planar graph uses 12-sided rectilinear polygons and takes O(nlogn) time. We describe a new algorithm that guarantees 10-sided rectilinear polygons and runs in O(n) time. We also describe a linear-time algorithm for proportional contact representation of planar 3-trees with 8-sided rectilinear polygons and show that this is optimal, as there exist planar 3-trees that require 8-sided polygons. Finally, we show that a maximal outer-planar graph admits a proportional contact representation using rectilinear polygons with 6 sides when the outer-boundary is a rectangle and with 4 sides otherwise.

Keywords

Planar Graph Simple Polygon Contact Representation Contact Graph Outer Vertex 
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 2011

Authors and Affiliations

  • Muhammad Jawaherul Alam
    • 1
  • Therese Biedl
    • 2
  • Stefan Felsner
    • 3
  • Andreas Gerasch
    • 4
  • Michael Kaufmann
    • 4
  • Stephen G. Kobourov
    • 1
  1. 1.Department of Computer ScienceUniversity of ArizonaTucsonUSA
  2. 2.David R. Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada
  3. 3.Institut für MathematikTechnische Universität BerlinBerlinGermany
  4. 4.Wilhelm-Schickhard-Institut für InformatikTübingen UniversitätTübingenGermany

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