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36th European Workshop on Computational Geometry Disjoint tree-compatible plane perfect matchings Oswin Aichholzer 1 , Julia Obmann 1 , Pavel Pat ak 2 , Daniel Perz 1 , and Josef Tkadlec 2 1 Graz University of Technology, Austria 2 IST Austria,

  1. 36th European Workshop on Computational Geometry Disjoint tree-compatible plane perfect matchings Oswin Aichholzer 1 , Julia Obmann 1 , Pavel Pat´ ak 2 , Daniel Perz 1 , and Josef Tkadlec 2 1 Graz University of Technology, Austria 2 IST Austria, Klosterneuburg, Austria 1 Julia Obmann Disjoint tree-compatible plane perfect matchings

  2. Compatibility of graphs Setting: set S of 2 n points in the plane in general position 2 i Julia Obmann Disjoint tree-compatible plane perfect matchings

  3. Compatibility of graphs Setting: set S of 2 n points in the plane in general position 2 ii Julia Obmann Disjoint tree-compatible plane perfect matchings

  4. Compatibility of graphs Setting: set S of 2 n points in the plane in general position 2 iii Julia Obmann Disjoint tree-compatible plane perfect matchings

  5. Compatibility of graphs Setting: set S of 2 n points in the plane in general position 2 iv Julia Obmann Disjoint tree-compatible plane perfect matchings

  6. Compatibility of graphs Setting: set S of 2 n points in the plane in general position compatible 2 v Julia Obmann Disjoint tree-compatible plane perfect matchings

  7. Compatibility of graphs Setting: set S of 2 n points in the plane in general position 2 vi Julia Obmann Disjoint tree-compatible plane perfect matchings

  8. Compatibility of graphs Setting: set S of 2 n points in the plane in general position 2 vii Julia Obmann Disjoint tree-compatible plane perfect matchings

  9. Compatibility of graphs Setting: set S of 2 n points in the plane in general position disjoint compatible 2 viii Julia Obmann Disjoint tree-compatible plane perfect matchings

  10. Compatibility of matchings 3 i Julia Obmann Disjoint tree-compatible plane perfect matchings

  11. Compatibility of matchings • compatibility graph: ◦ vertices: all plane perfect matchings on S ◦ edge ( M i , M j ) ⇐ ⇒ M i and M j are compatible 3 ii Julia Obmann Disjoint tree-compatible plane perfect matchings

  12. Compatibility of matchings • compatibility graph: ◦ vertices: all plane perfect matchings on S ◦ edge ( M i , M j ) ⇐ ⇒ M i and M j are compatible • compatibility graph for matchings is connected convex point set: [C. Hernando, F. Hurtado and M. Noy; 2002.] general point set: [M.E. Houle, F. Hurtado, M. Noy and E. Rivera-Campo; 2005.] 3 iii Julia Obmann Disjoint tree-compatible plane perfect matchings

  13. Compatibility of matchings • compatibility graph: ◦ vertices: all plane perfect matchings on S ◦ edge ( M i , M j ) ⇐ ⇒ M i and M j are compatible • compatibility graph for matchings is connected convex point set: [C. Hernando, F. Hurtado and M. Noy; 2002.] general point set: [M.E. Houle, F. Hurtado, M. Noy and E. Rivera-Campo; 2005.] • diameter is O (log n ) [ABDGHHKMRSSUW; 2009.] and Ω(log n/ log log n ) [A.Razen; 2008.] 3 iv Julia Obmann Disjoint tree-compatible plane perfect matchings

  14. Disjoint compatibility of matchings 4 i Julia Obmann Disjoint tree-compatible plane perfect matchings

  15. Disjoint compatibility of matchings • disjoint compatibility graph: ◦ vertices: all plane perfect matchings on S ◦ edge ( M i , M j ) ⇐ ⇒ M i , M j are disjoint compatible 4 ii Julia Obmann Disjoint tree-compatible plane perfect matchings

  16. Disjoint compatibility of matchings • disjoint compatibility graph: ◦ vertices: all plane perfect matchings on S ◦ edge ( M i , M j ) ⇐ ⇒ M i , M j are disjoint compatible • disjoint compatibility graph for matchings on point sets of 2 n ≥ 6 points in convex position is disconnected [O. Aichholzer, A. Asinowski and T. Miltzow; 2015.] 4 iii Julia Obmann Disjoint tree-compatible plane perfect matchings

  17. Disjoint compatibility of matchings • disjoint compatibility graph: ◦ vertices: all plane perfect matchings on S ◦ edge ( M i , M j ) ⇐ ⇒ M i , M j are disjoint compatible • disjoint compatibility graph for matchings on point sets of 2 n ≥ 6 points in convex position is disconnected [O. Aichholzer, A. Asinowski and T. Miltzow; 2015.] Alternative way of defining compatibility? 4 iv Julia Obmann Disjoint tree-compatible plane perfect matchings

  18. Disjoint tree-compatibility of matchings 5 i Julia Obmann Disjoint tree-compatible plane perfect matchings

  19. Disjoint tree-compatibility of matchings • consider ’compatibility’ via disjoint compatible plane spanning trees 5 ii Julia Obmann Disjoint tree-compatible plane perfect matchings

  20. Disjoint tree-compatibility of matchings • consider ’compatibility’ via disjoint compatible plane spanning trees 5 iii Julia Obmann Disjoint tree-compatible plane perfect matchings

  21. Disjoint tree-compatibility of matchings • consider ’compatibility’ via disjoint compatible plane spanning trees 5 iv Julia Obmann Disjoint tree-compatible plane perfect matchings

  22. Disjoint tree-compatibility of matchings • consider ’compatibility’ via disjoint compatible plane spanning trees not compatible! 5 v Julia Obmann Disjoint tree-compatible plane perfect matchings

  23. Disjoint tree-compatibility of matchings • consider ’compatibility’ via disjoint compatible plane spanning trees 5 vi Julia Obmann Disjoint tree-compatible plane perfect matchings

  24. Disjoint tree-compatibility of matchings • consider ’compatibility’ via disjoint compatible plane spanning trees 5 vii Julia Obmann Disjoint tree-compatible plane perfect matchings

  25. Disjoint tree-compatibility of matchings • consider ’compatibility’ via disjoint compatible plane spanning trees matchings are disjoint tree-compatible (for short: tree-compatible) 5 viii Julia Obmann Disjoint tree-compatible plane perfect matchings

  26. Disjoint tree-compatibility of matchings • consider ’compatibility’ via disjoint compatible plane spanning trees • disjoint tree-compatibility graph G 2 n : ◦ vertices: all plane perfect matchings on S ◦ edge ( M i , M j ) ⇐ ⇒ M i , M j disjoint tree-compatible 5 ix Julia Obmann Disjoint tree-compatible plane perfect matchings

  27. Disjoint tree-compatibility of matchings • consider ’compatibility’ via disjoint compatible plane spanning trees ATTENTION: different from (disjoint) compatibility! • disjoint tree-compatibility graph G 2 n : disjoint tree-compatible � compatible ◦ vertices: all plane perfect matchings on S disjoint compatible � disjoint tree-compatible ◦ edge ( M i , M j ) ⇐ ⇒ M i , M j disjoint tree-compatible 5 x Julia Obmann Disjoint tree-compatible plane perfect matchings

  28. Disjoint tree-compatibility of matchings • consider ’compatibility’ via disjoint compatible plane spanning trees disjoint compatible NOT disjoint tree-compatible ATTENTION: different from (disjoint) compatibility! disjoint tree-compatible � compatible disjoint compatible � disjoint tree-compatible 5 xi Julia Obmann Disjoint tree-compatible plane perfect matchings

  29. G 8 6 i Julia Obmann Disjoint tree-compatible plane perfect matchings

  30. G 8 6 ii Julia Obmann Disjoint tree-compatible plane perfect matchings

  31. G 8 6 iii Julia Obmann Disjoint tree-compatible plane perfect matchings

  32. G 8 6 iv Julia Obmann Disjoint tree-compatible plane perfect matchings

  33. G 8 6 v Julia Obmann Disjoint tree-compatible plane perfect matchings

  34. G 8 6 vi Julia Obmann Disjoint tree-compatible plane perfect matchings

  35. G 8 6 vii Julia Obmann Disjoint tree-compatible plane perfect matchings

  36. G 8 6 viii Julia Obmann Disjoint tree-compatible plane perfect matchings

  37. G 8 6 ix Julia Obmann Disjoint tree-compatible plane perfect matchings

  38. Upper bound 7 i Julia Obmann Disjoint tree-compatible plane perfect matchings

  39. Upper bound Theorem 1. For 2 n ≥ 10 , the graph G 2 n is connected and diam( G 2 n ) ≤ 5 . 7 ii Julia Obmann Disjoint tree-compatible plane perfect matchings

  40. Upper bound Theorem 1. For 2 n ≥ 10 , the graph G 2 n is connected and diam( G 2 n ) ≤ 5 . Proof Idea. 7 iii Julia Obmann Disjoint tree-compatible plane perfect matchings

  41. Upper bound Theorem 1. For 2 n ≥ 10 , the graph G 2 n is connected and diam( G 2 n ) ≤ 5 . Proof Idea. • ”inside semicycles” can be (simultaneously) rotated in one step 7 iv Julia Obmann Disjoint tree-compatible plane perfect matchings

  42. Upper bound Theorem 1. For 2 n ≥ 10 , the graph G 2 n is connected and diam( G 2 n ) ≤ 5 . Proof Idea. • ”inside semicycles” can be (simultaneously) rotated in one step X 1 X 2 7 v Julia Obmann Disjoint tree-compatible plane perfect matchings

  43. Upper bound Theorem 1. For 2 n ≥ 10 , the graph G 2 n is connected and diam( G 2 n ) ≤ 5 . Proof Idea. • ”inside semicycles” can be (simultaneously) rotated in one step X 1 X 2 7 vi Julia Obmann Disjoint tree-compatible plane perfect matchings

  44. Upper bound Theorem 1. For 2 n ≥ 10 , the graph G 2 n is connected and diam( G 2 n ) ≤ 5 . Proof Idea. • ”inside semicycles” can be (simultaneously) rotated in one step 7 vii Julia Obmann Disjoint tree-compatible plane perfect matchings

  45. Upper bound Theorem 1. For 2 n ≥ 10 , the graph G 2 n is connected and diam( G 2 n ) ≤ 5 . Proof Idea. • ”inside semicycles” can be (simultaneously) rotated in one step • large ”semiears” ( ≥ 12 vertices) can be rotated in at most 3 steps 7 viii Julia Obmann Disjoint tree-compatible plane perfect matchings

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