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Convex partitions with 2-edge connected dual graphs

Author

Listed:
  • Marwan Al-Jubeh

    (Tufts University)

  • Michael Hoffmann

    (ETH Zürich)

  • Mashhood Ishaque

    (Tufts University)

  • Diane L. Souvaine

    (Tufts University)

  • Csaba D. Tóth

    (University of Calgary)

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 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.

Suggested Citation

  • Marwan Al-Jubeh & Michael Hoffmann & Mashhood Ishaque & Diane L. Souvaine & Csaba D. Tóth, 2011. "Convex partitions with 2-edge connected dual graphs," Journal of Combinatorial Optimization, Springer, vol. 22(3), pages 409-425, October.
    • Handle: RePEc:spr:jcomop:v:22:y:2011:i:3:d:10.1007_s10878-010-9310-1
    DOI: 10.1007/s10878-010-9310-1
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