Convex Partitions with 2-Edge Connected Dual Graphs

  • Marwan Al-Jubeh
  • Michael Hoffmann
  • Mashhood Ishaque
  • Diane L. Souvaine
  • Csaba D. Tóth
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5609)

Abstract

It is shown that for every finite set of disjoint convex polygonal obstacles in the plane, with a total of n vertices, the free space around the obstacles can be partitioned into open convex cells whose dual graph (defined below) is 2-edge connected. Intuitively, every edge of the dual graph corresponds to a pair of adjacent cells that are both incident to the same vertex.

Aichholzer et al. recently conjectured that given an even number of line-segment obstacles, one can construct a convex partition by successively extending the segments along their supporting lines such that the dual graph is the union of two edge-disjoint spanning trees. Here we present a counterexamples to this conjecture, with n disjoint line segments for any n ≥ 15, such that the dual graph of any convex partition constructed by this method has a bridge edge, and thus the dual graph cannot be partitioned into two spanning trees.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Marwan Al-Jubeh
    • 1
  • Michael Hoffmann
    • 2
  • Mashhood Ishaque
    • 1
  • Diane L. Souvaine
    • 1
  • Csaba D. Tóth
    • 3
  1. 1.Department of Computer ScienceTufts UniversityMefordUSA
  2. 2.Institute of Theoretical Computer ScienceETH ZürichSwitzerland
  3. 3.Department of MathematicsUniversity of CalgaryCanada

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