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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Al-Jubeh, M., Hoffmann, M., Ishaque, M., Souvaine, D.L., Tóth, C.D. (2009). Convex Partitions with 2-Edge Connected Dual Graphs. In: Ngo, H.Q. (eds) Computing and Combinatorics. COCOON 2009. Lecture Notes in Computer Science, vol 5609. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02882-3_20
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DOI: https://doi.org/10.1007/978-3-642-02882-3_20
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