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1 Drawing Tree-Based Phylogenetic Networks with Minimum Number of Crossings Jonathan Klawitter Peter Stumpf GD 2020 2 - 1 Phylogenetic networks T 1 2 3 4 5 2 - 2 Phylogenetic networks T N 1 2 3 4 5 1 2 3 4 5 2 - 3

  1. 1 Drawing Tree-Based Phylogenetic Networks with Minimum Number of Crossings Jonathan Klawitter · Peter Stumpf GD 2020

  2. 2 - 1 Phylogenetic networks T 1 2 3 4 5

  3. 2 - 2 Phylogenetic networks T N 1 2 3 4 5 1 2 3 4 5

  4. 2 - 3 Phylogenetic networks T N N 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

  5. 2 - 4 Phylogenetic networks T N T ′ N 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

  6. 3 - 1 Kumar et al. “The evolutionary history of bears is characterized by gene flow across species”

  7. 3 - 2 Kumar et al. “The evolutionary history of bears is Richards et al. “Phylogenomic Analysis Demonstrates a characterized by gene flow across species” Pattern of Rare and Ancient Horizontal Gene Transfer between Plants and Fungi”

  8. 3 - 3 Kumar et al. “The evolutionary history of bears is Richards et al. “Phylogenomic Analysis Demonstrates a characterized by gene flow across species” Pattern of Rare and Ancient Horizontal Gene Transfer between Plants and Fungi” Vaughan et al. “Inferring Ancestral Recombination Graphs from Bacterial Genomic Data”

  9. 4 - 1 Drawing styles

  10. 4 - 2 Drawing styles

  11. 4 - 3 Drawing styles

  12. 4 - 4 Drawing styles

  13. 4 - 5 Drawing styles

  14. 4 - 6 Drawing styles snake style

  15. 4 - 7 Drawing styles snake style

  16. 4 - 8 Drawing styles snake style ear style

  17. 4 - 9 Drawing styles snake style ear style horizontal style

  18. 4 - 10 Drawing styles snake style ear style horizontal style

  19. 4 - 11 Drawing styles snake style ear style horizontal style

  20. 5 - 1 Let N be a tree-based network on n leaves and k reticulation edges . A crossing-minimal ear-style drawing of N can be computed in O ( nk ) time.

  21. 5 - 2 Let N be a tree-based network on n leaves and k reticulation edges . A crossing-minimal ear-style drawing of N can be computed in O ( nk ) time.

  22. 5 - 3 Let N be a tree-based network on n leaves and k reticulation edges . A crossing-minimal ear-style drawing of N can be computed in O ( nk ) time.

  23. 5 - 4 Let N be a tree-based network on n leaves and k reticulation edges . A crossing-minimal ear-style drawing of N can be computed in O ( nk ) time.

  24. 5 - 5 Let N be a tree-based network on n leaves and k reticulation edges . A crossing-minimal ear-style drawing of N can be computed in O ( nk ) time.

  25. 5 - 6 Let N be a tree-based network on n leaves and k reticulation edges . A crossing-minimal ear-style drawing of N can be computed in O ( nk ) time.

  26. 6 - 1 Ear-style algorithm

  27. 6 - 2 Ear-style algorithm 0 0 0 0 0 0 0 0

  28. 6 - 3 Ear-style algorithm 0 0 0 0 0 0 0 0

  29. 6 - 4 Ear-style algorithm 0 0 0 0 0 0 0 0

  30. 6 - 5 Ear-style algorithm 0 0 0 0 0 0 0 1

  31. 6 - 6 Ear-style algorithm 2 0 0 2 0 0 0 1

  32. 6 - 7 Ear-style algorithm 2 3 0 0 2 0 1 1

  33. 6 - 8 Ear-style algorithm 2 3 0 0 2 0 1 1

  34. 6 - 9 Ear-style algorithm 3 3 0 1 2 1

  35. 6 - 10 Ear-style algorithm 3 3 0 1 2 1

  36. 6 - 11 Ear-style algorithm 3 3 1 0 2 1

  37. 6 - 12 Ear-style algorithm 3 3 2 1

  38. 6 - 13 Ear-style algorithm 3 3 2 1

  39. 6 - 14 Ear-style algorithm 3 4

  40. 6 - 15 Ear-style algorithm 3 4

  41. 6 - 16 Ear-style algorithm 4 3

  42. 6 - 17 Ear-style algorithm

  43. 7 - 1 The Crossing Minimisation problem for horizontal-style drawings of tree-based networks is NP-complete.

  44. 7 - 2 The Crossing Minimisation problem for horizontal-style drawings of tree-based networks is NP-complete. G MaxCut

  45. 8 - 1 Horizontal style NP-reduction G u e g w v f

  46. 8 - 2 Horizontal style NP-reduction v e u e e ′ N e G u e g w v f

  47. 8 - 3 Horizontal style NP-reduction w g v e u g u e e ′ g ′ N e N g G u e g w v f

  48. 8 - 4 Horizontal style NP-reduction w g w f v f v e u g u e e ′ g ′ f ′ N e N g N f G u e g w v f

  49. 8 - 5 Horizontal style NP-reduction w g w f v f v e u g u e e ′ g ′ f ′ N e N g N f G u e g w v f

  50. 8 - 6 Horizontal style NP-reduction w g w f v f v e u g u g u e u e e ′ g ′ f ′ N e N g N f G u e g w v f

  51. 8 - 7 Horizontal style NP-reduction u ⋆ w g w f v f v e u g u g u e u e e ′ g ′ f ′ N e N g N f G u e g w v f

  52. 8 - 8 Horizontal style NP-reduction u ⋆ w g w f v f v e u g u g u e u e e ′ g ′ f ′ N e N g N f G u e g w v f

  53. 8 - 9 Horizontal style NP-reduction u ⋆ w g w f v f v e u g u g u e u e e ′ g ′ f ′ N e N g N f G u e g w v f

  54. 8 - 10 Horizontal style NP-reduction G u e g w v f

  55. 8 - 11 Horizontal style NP-reduction G u e g w v f

  56. 8 - 12 Horizontal style NP-reduction G u e g w v f

  57. 8 - 13 Horizontal style NP-reduction G u e g w v f

  58. 8 - 14 Horizontal style NP-reduction G u e g w v f

  59. 8 - 15 Horizontal style NP-reduction G u e g w v f

  60. 9 - 1 Let N be a tree-based network on n leaves and k reticulation edges . A crossing-minimal snake-style drawing of N can be computed in O ( 2 k nk ) time.

  61. 9 - 2 Let N be a tree-based network on n leaves and k reticulation edges . A crossing-minimal snake-style drawing of N can be computed in O ( 2 k nk ) time.

  62. 9 - 3 Let N be a tree-based network on n leaves and k reticulation edges . A crossing-minimal snake-style drawing of N can be computed in O ( 2 k nk ) time.

  63. 10 Let N be a tree-based network on n leaves and k reticulation edges . A crossing-minimal ear-style drawing of N can be computed in O ( nk ) time. The Crossing Minimisation problem for horizontal-style drawings of tree-based networks is NP-complete. Let N be a tree-based network on n leaves and k reticulation edges . A crossing-minimal snake-style drawing of N can be computed in O ( 2 k nk ) time.

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