New results on a visibility representation of graphs in 3D

  • Sándor P. Fekete
  • Michael E. Houle
  • Sue Whitesides
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1027)

Abstract

This paper considers a 3-dimensional visibility representation of cliques Kn. In this representation, the objects representing the vertices are 2-dimensional and lie parallel to the x, y-plane, and two vertices of the graph are adjacent if and only if their corresponding objects see each other by a line of sight parallel to the z-axis that intersects the interiors of the objects. In particular, we represent vertices by unit discs and by discs of arbitrary radii (possibly different for different vertices); we also represent vertices by axis-aligned unit squares, by axis-aligned squares of arbitrary size (possibly different for different vertices), and by axis-aligned rectangles.

We present:
  • a significant improvement (from 102 to 55) of the best known upper bound for the size of cliques representable by rectangles or squares of arbitrary size;

  • a sharp bound for the representation of cliques by unit squares (K7 can be represented but Kn for n>7 cannot);

  • a representation of Kn by unit discs.

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Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Sándor P. Fekete
    • 1
  • Michael E. Houle
    • 2
  • Sue Whitesides
    • 3
  1. 1.Center for Parallel ComputingUniversität zu KölnKölnGermany
  2. 2.Department of Computer ScienceUniversity of NewcastleCallaghanAustralia
  3. 3.School of Computer ScienceMcGill UniversityMontréalCanada

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