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

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

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 ]

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

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]

6 Reconfiguration of Geometric Matchings

7

8

9

10 Reconfiguration of Geometric Matchings

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]

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

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

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.

15 (Straight-Forward) Convex Partition

16 (Straight-Forward) Convex Partition

17 (Straight-Forward) Convex Partition

18 (Straight-Forward) Convex Partition

19 (Straight-Forward) Convex Partition

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

21 Dual Graph of the Convex Partition

22 Dual Graph of the Convex Partition

23 Dual Graph of the Convex Partition

24 Dual Graph of the Convex Partition

25 Dual Graph of the Convex Partition

26 Dual Graph of the Convex Partition

27 Dual Graph of the Convex Partition

28 Dual Graph of the Convex Partition

29 Dual Graph of the Convex Partition

30 Dual Graph of the Convex Partition

31 Dual Graph of the Convex Partition

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

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 ]

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 ]

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

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

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

38

39 A3 A1 A4 A5A2 Cut-Edge

40 A3 A1 A4 A5A2 Cut-Edge

41

42 B3 B1 B4 B5B2 Cut-Edge

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

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?

45 More General Convex Partitions

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

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

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.

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.

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.

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.

52 Thank You.