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1 Spaces of phylogenetic networks Jonathan Klawitter PhD Exam 5th March, 2020 2 - 1 Phylogenetic trees & networks 2 - 2 Phylogenetic trees & networks 2 - 3 Phylogenetic trees & networks T 1 2 3 4 5 6 2 - 4

  1. 1 Spaces of phylogenetic networks Jonathan Klawitter PhD Exam · 5th March, 2020

  2. 2 - 1 Phylogenetic trees & networks

  3. 2 - 2 Phylogenetic trees & networks

  4. 2 - 3 Phylogenetic trees & networks T 1 2 3 4 5 6

  5. 2 - 4 Phylogenetic trees & networks T N 1 2 3 4 5 6 1 2 3 4 5 6

  6. 3 - 1 Set of networks 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 4 3 2 1 3 4 1 4 2 3 1 3 2 4 1 3 2 4 1 3 2 4 1 2 4 3 2 1 3 4 1 4 2 3 2 1 4 3 . . . 1 2 3 4 1 2 4 3 1 3 2 4 1 3 4 2 1 2 3 4 1 2 4 3 1 3 4 2 . . . 1 2 3 4 1 2 3 4 1 2 4 3 1 2 4 3 1 2 3 4 1 2 3 4 1 2 3 4 . . . 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

  7. 3 - 2 Set of networks 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 4 3 2 1 3 4 1 4 2 3 trees 1 3 2 4 1 3 2 4 1 3 2 4 1 2 4 3 2 1 3 4 1 4 2 3 2 1 4 3 . . . 1 2 3 4 1 2 4 3 1 3 2 4 1 3 4 2 1 2 3 4 1 2 4 3 1 3 4 2 . . . 1 2 3 4 1 2 3 4 1 2 4 3 1 2 4 3 1 2 3 4 1 2 3 4 1 2 3 4 . . . 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

  8. 3 - 3 Set of networks 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 4 3 2 1 3 4 1 4 2 3 trees 1 3 2 4 1 3 2 4 1 3 2 4 1 2 4 3 2 1 3 4 1 4 2 3 2 1 4 3 . . . normal 1 2 3 4 1 2 4 3 1 3 2 4 1 3 4 2 1 2 3 4 1 2 4 3 1 3 4 2 . . . 1 2 3 4 1 2 3 4 1 2 4 3 1 2 4 3 1 2 3 4 1 2 3 4 1 2 3 4 . . . 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

  9. 3 - 4 Set of networks 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 4 3 2 1 3 4 1 4 2 3 trees 1 3 2 4 1 3 2 4 1 3 2 4 1 2 4 3 2 1 3 4 1 4 2 3 2 1 4 3 . . . normal 1 2 3 4 1 2 4 3 1 3 2 4 1 3 4 2 1 2 3 4 1 2 4 3 1 3 4 2 . . . tree- child 1 2 3 4 1 2 3 4 1 2 4 3 1 2 4 3 1 2 3 4 1 2 3 4 1 2 3 4 . . . 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

  10. 3 - 5 Set of networks 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 4 3 2 1 3 4 1 4 2 3 trees 1 3 2 4 1 3 2 4 1 3 2 4 1 2 4 3 2 1 3 4 1 4 2 3 2 1 4 3 . . . normal 1 2 3 4 1 2 4 3 1 3 2 4 1 3 4 2 1 2 3 4 1 2 4 3 1 3 4 2 . . . tree- child 1 2 3 4 1 2 3 4 1 2 4 3 1 2 4 3 1 2 3 4 1 2 3 4 1 2 3 4 . . . 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 tree-based . . .

  11. 4 - 1 Rearrangement operations NNI u u . . . z x x x y y z x y y x y x y

  12. 4 - 2 Rearrangement operations NNI u u . . . z x x x y y z x y y x y x y SNPR 1 2 3 4 1 2 3 4

  13. 4 - 3 Rearrangement operations NNI u u . . . z x x x y y z x y y x y x y SNPR 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

  14. 4 - 4 Rearrangement operations NNI u u . . . z x x x y y z x y y x y x y SNPR 1 2 3 4 1 2 3 4 PR 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

  15. 5 NNI tree space 1 2 3 4 1 2 3 4 1 2 4 3 1 2 3 4 2 1 3 4 1 3 2 4 2 1 4 3 2 1 3 4 1 4 2 3 1 2 4 3 1 2 3 4 1 3 2 4 1 4 2 3 1 3 2 4 1 2 3 4

  16. 6 - 1 Chapters

  17. 6 - 2 Chapters � Connectedness & diameter

  18. 6 - 3 Chapters � Connectedness & diameter � Neighbourhood size

  19. 6 - 4 Chapters � Connectedness & diameter � Neighbourhood size � SNPR- and PR-distance

  20. 6 - 5 Chapters � Connectedness & diameter � Neighbourhood size � SNPR- and PR-distance � Maximum agreement graphs

  21. 7 - 1 Connectedness & diameter

  22. 7 - 2 Connectedness & diameter

  23. 7 - 3 Connectedness & diameter

  24. 8 - 1 Neighbourhood 3 3 3 3 2 2 2 2 2 2 3 1 1 1 1 1 1 3 N 2 3 2 3 2 2 3 3 3 2 3 1 2 3 1 2 1 1 1 2 1 3 1 1 1 2 3 1 2 3

  25. 8 - 2 Neighbourhood � Methods � Count number of possible operations � Subtract number of trivial operations � Correct for double counting

  26. 8 - 3 Neighbourhood � Methods � Count number of possible operations � Subtract number of trivial operations � Correct for double counting � Results Neighbourhood size for � trees under NNI and SNPR � tree-child networks under NNI and SNPR � normal networks under SNPR � general networks (bounds)

  27. 9 - 1 PR-distance Theorem. Computing the SPR-distance of two trees is NP-hard.

  28. 9 - 2 PR-distance Theorem. Computing the SPR-distance of two trees is NP-hard. Theorem. The space of trees embeds isometrically into the space of networks unter SNPR and PR. Corollary. Computing the SNPR- and PR-distance of two networks is NP-hard.

  29. 9 - 3 PR-distance Theorem. Computing the SPR-distance of two trees is NP-hard. Theorem. The space of trees embeds isometrically into the space of networks unter SNPR and PR. Corollary. Computing the SNPR- and PR-distance of two networks is NP-hard. Theorem. Let N have r reticulations. T ′ ∈ D ( N ) d PR ( T , T ′ ) + r d PR ( T , N ) = min

  30. 9 - 4 PR-distance Theorem. Computing the SPR-distance of two trees is NP-hard. Theorem. The space of trees embeds isometrically into the space of networks unter SNPR and PR. Corollary. Computing the SNPR- and PR-distance of two networks is NP-hard. Theorem. Let N have r reticulations. T ′ ∈ D ( N ) d PR ( T , T ′ ) + r d PR ( T , N ) = min Corollary. Computing d PR ( T , N ) is fixed-parameter tractable.

  31. 10 - 1 Maximum agreement forests T ′ T ρ ρ ρ 1 2 3 4 5 1 4 2 3 5 4 1 2 3 5

  32. 10 - 2 Maximum agreement forests T ′ T ρ ρ ρ 1 2 3 4 5 1 4 2 3 5 4 1 2 3 5 ρ 1 2 3 4 5

  33. 10 - 3 Maximum agreement forests T ′ T ρ ρ ρ 1 2 3 4 5 1 4 2 3 5 4 1 2 3 5 ρ ρ 1 2 3 4 5 1 2 3 4 5

  34. 10 - 4 Maximum agreement forests T ′ T ρ ρ ρ 1 2 3 4 5 1 4 2 3 5 4 1 2 3 5 F ρ 1 5 2 3 4

  35. 10 - 5 Maximum agreement forests T ′ T ρ ρ ρ 1 2 3 4 5 1 4 2 3 5 4 1 2 3 5 F ρ ρ 1 5 2 3 4 1 2 3 4 5

  36. 10 - 6 Maximum agreement forests T ′ T ρ ρ ρ 1 2 3 4 5 1 4 2 3 5 4 1 2 3 5 F ρ ρ ρ 1 5 2 3 4 1 2 3 4 5 4 1 2 3 5

  37. 10 - 7 Maximum agreement forests T ′ T ρ ρ ρ 1 2 3 4 5 1 4 2 3 5 4 1 2 3 5 Theorem. The size of a maximum agreement forest for two trees characterises their SPR-distance. F ρ ρ ρ 1 5 2 3 4 1 2 3 4 5 4 1 2 3 5

  38. 11 - 1 Maximum agreement graphs N ′ N ρ ρ ρ 1 2 3 4 5 1 4 2 3 5 1 4 2 3 5

  39. 11 - 2 Maximum agreement graphs N ′ N ρ ρ ρ 1 2 3 4 5 1 4 2 3 5 1 4 2 3 5 ρ 1 4 5 2 3

  40. 11 - 3 Maximum agreement graphs N ′ N ρ ρ ρ 1 2 3 4 5 1 4 2 3 5 1 4 2 3 5 G ρ 1 4 5 2 3

  41. 11 - 4 Maximum agreement graphs N ′ N ρ ρ ρ 1 2 3 4 5 1 4 2 3 5 1 4 2 3 5 G ρ ρ 1 4 5 2 3 1 2 3 4 5

  42. 11 - 5 Maximum agreement graphs N ′ N ρ ρ ρ 1 2 3 4 5 1 4 2 3 5 1 4 2 3 5 G ρ ρ ρ 1 4 5 2 3 1 2 3 4 5 1 4 2 3 5

  43. 12 - 1 Agreement distance Theorem. The number of sprouts and disagreement edges of a maximum agreement graph define a metric, the agreement distance d AD .

  44. 12 - 2 Agreement distance Theorem. The number of sprouts and disagreement edges of a maximum agreement graph define a metric, the agreement distance d AD . Theorem. d AD ( T , N ) = d PR ( T , N )

  45. 12 - 3 Agreement distance Theorem. The number of sprouts and disagreement edges of a maximum agreement graph define a metric, the agreement distance d AD . Theorem. d AD ( T , N ) = d PR ( T , N ) Theorem. d AD ( N , N ′ ) ≤ d PR ( N , N ′ ) ≤ 3 d AD ( N , N ′ ) d AD ( N , N ′ ) ≤ d SNPR ( N , N ′ ) ≤ 6 d AD ( N , N ′ )

  46. 13 - 1 Publications � K., “The SNPR-Neighbourhood of tree-child networks”, Journal of Graph Algorithms & Applications, 2018. � K., Linz, “On the Subnet Prune and Regraft Distance”, Electronic Journal of Combinatorics, 2019. � K., “The agreement distance of rooted phylogenetic networks ”, Discrete Mathematics & Theoretical Computer Science, 2019.

  47. 13 - 2 Publications � K., “The SNPR-Neighbourhood of tree-child networks”, Journal of Graph Algorithms & Applications, 2018. � K., Linz, “On the Subnet Prune and Regraft Distance”, Electronic Journal of Combinatorics, 2019. � K., “The agreement distance of rooted phylogenetic networks ”, Discrete Mathematics & Theoretical Computer Science, 2019. � Janssen, K., “Rearrangement operations on unrooted phylogenetic networks ”, Theory and Applications of Graphs, 2019. � K., “The agreement distance of unrooted phylogenetic networks ”, Discrete Mathematics & Theoretical Computer Science, 2020 (to appear).

  48. 14

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