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Convex Partitions with 2-Edge Connected Dual Graphs Marwan Al-JubehMichael Hoffmann Diane L. SouvaineCsaba D. Toth 15th International Computing and Combinatorics Conference Mashhood Ishaque

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2 Outline for the talk –Reconfiguration of geometric objects –Reconfiguration of geometric matchings –Tools: convex partitions and dual graphs –The two spanning trees conjecture for geometric matchings Our results –Refuting the two spanning trees conjecture –Repairing the conjecture

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3 Reconfiguration of Geometric Objects Transform one geometric object into another using a sequence of allowed moves. Any triangulation on a planar point set can be transformed into any other triangulation using “diagonal edge-flips”. [ Osherovich and Bruckstein, 2007 ] Any non-crossing spanning tree can be transformed into any other non-crossing spanning tree in O(log n) moves (a move replaces a non-crossing spanning with another non-crossing one). [ Aichholzer et al., 2002 ]

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4 Matching (set of line segments) Perfect Matching Non-Crossing Matching Compatible Matchings (union is non-crossing) Disjoint Matchings (no edge is repeated) Disjoint Compatible Matchings Reconfiguration of Geometric Matchings

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5 Given two perfect matchings M 1 and M 2 on the same set of 2n points (in general position) in the plane, reconfigure M 1 into M 2 through a sequence of moves. Each move can replace a matching by a compatible matching. Can be done in O(log n) moves [ Aichholzer et al., 2007 ] Sometimes takes Ω(log n/ log log n) moves [ Razen, 2008 ] Close the gap between the lower and the upper bounds. [Open]

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6 Reconfiguration of Geometric Matchings

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10 Reconfiguration of Geometric Matchings

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11 Disjoint Compatible Matchings? 1.Given a perfect matching M on 4n points (in general position) in the plane, is there a disjoint compatible matching? [Open] 2.Can any matching M 1 be reconfigured to another matching M 2 via disjoint compatible matching moves? [Open] 3.What is the maximal diameter of the graph of compatible disjoint matchings on 4n points in general position? [Open]

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12 Disjoint Compatible Matching? (Odd Case) For any odd number of line segments, there are examples where no compatible disjoint matching exists.

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13 Outline for the talk –Reconfiguration of geometric objects –Reconfiguration of geometric matchings –Tools: convex partitions and dual graphs –The two spanning trees conjecture for geometric matchings Our results –Refuting the two spanning trees conjecture –Repairing the conjecture

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14 (Straight-Forward) Convex Partition Extend each segment in a straight-line until the extension hits another segment, a previous extension or the bounding box.

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15 (Straight-Forward) Convex Partition

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16 (Straight-Forward) Convex Partition

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17 (Straight-Forward) Convex Partition

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18 (Straight-Forward) Convex Partition

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19 (Straight-Forward) Convex Partition

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20 (Straight-Forward) Convex Partition n segments, (2n)! possible straight-forward convex partitions

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21 Dual Graph of the Convex Partition

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22 Dual Graph of the Convex Partition

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23 Dual Graph of the Convex Partition

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24 Dual Graph of the Convex Partition

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25 Dual Graph of the Convex Partition

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26 Dual Graph of the Convex Partition

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27 Dual Graph of the Convex Partition

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28 Dual Graph of the Convex Partition

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29 Dual Graph of the Convex Partition

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30 Dual Graph of the Convex Partition

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31 Dual Graph of the Convex Partition

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32 Outline for the talk –Reconfiguration of geometric objects –Reconfiguration of geometric matchings –Tools: convex partitions and dual graphs –The two spanning trees conjecture for geometric matchings Our results –Refuting the two spanning trees conjecture –Repairing the conjecture

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33 Two Spanning Trees Conjecture For n disjoint line segments in general position, there is a (straight-forward) convex partition such that the dual graph is the edge-disjoint union of two spanning trees, and the two endpoints of each segment correspond to different spanning trees. [ Aichholzer et al., 2007 ]

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34 Two Spanning Trees Conjecture For 2n segments, the conjecture implies that there is a compatible disjoint matching, where each edge connects points in a convex face. [ Aichholzer et al., 2007 ]

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35 Outline for the talk –Reconfiguration of geometric objects –Reconfiguration of geometric matchings –Tools: convex partitions and dual graphs –The two spanning trees conjecture for geometric matchings Our results –Refuting the two spanning trees conjecture –Repairing the conjecture

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36 Refuting the Conjecture –For a graph to contain a union of two edge-disjoint spanning trees, it must be 2-edge connected (necessary condition, but not sufficient) –A graph that is not 2-edge connected contains a cut-edge (bridge). Cut-Edge

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37 For every n 15, there is an arrangement of n disjoint line segments such that there is a cut-edge in the dual graph for any straight- forward convex partition. Refuting the Conjecture

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39 A3 A1 A4 A5A2 Cut-Edge

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40 A3 A1 A4 A5A2 Cut-Edge

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42 B3 B1 B4 B5B2 Cut-Edge

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43 Outline for the talk –Reconfiguration of geometric objects –Reconfiguration of geometric matchings –Tools: convex partitions and dual graphs –The two spanning trees conjecture for geometric matchings Our results –Refuting the two spanning trees conjecture –Repairing the conjecture

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44 Repairing the Conjecture Straight-forward convex partitions does not always has 2-edge connected dual graphs (hence no two edge-disjoint spanning trees), but what if we allow arbitrary extensions as long as we have a convex partition?

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45 More General Convex Partitions

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46 For any set of disjoint line segments in the plane in general position, there is a convex partition whose dual graph is 2-edge connected. 2-Edge Connected Dual Graphs

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47 Extended Path and Extension Trees Extension trees defined by [ Bose et al., 2000 ]

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48 Characterization of a Cut-Edge There is a cut-edge in the dual graph iff there is an extended-path that starts from the endpoint of a segment s, and ends at the same segment s.

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49 Fixing a Cut-Edge: Continuous Deformation of a Closed Curve We continuously deform every such cycle until one extension tree breaks into two trees.

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50 Fixing a Cut-Edge: Continuous Deformation of a Closed Curve Termination is guaranteed because: –Each extension tree is fixed without affecting the other extension trees. –Each time an extension tree is fixed, although its sub-trees might still be problematic but they are strictly smaller in size. (size = # of segment endpoint in the tree) –There are at most 2n extension trees, and an extension tree of size 1 can not contain a problematic extended path.

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51 Repairing the Two Spanning Trees Conjecture For n disjoint line segments in general position, there is a (straight-forward) convex partition such that the dual graph is the edge-disjoint union of two spanning trees, and the two endpoints of each segment correspond to different spanning trees.

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52 Thank You.

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