Convex Partitions with 2-Edge Connected Dual Graphs

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. Keywords Convex Partitions · Dual Graphs · Geometric Matchings

• Type: Conference
• Year: 2009
• Where: COCOON
• Authors: Marwan Al-Jubeh, Michael Hoffmann, Mashhood Ishaque, Diane L. Souvaine, Csaba D. Tóth
• Keywords: COCOON 2009, Combinatorics, Convex Partitions, Convex Polygonal Obstacles, Dual Graphs