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Quaret Compatibility: New Results and Surprising Counterexamples

Quaret Compatibility: New Results and Surprising Counterexamples. Stefan Gr ünewald. Compatibility Problem. Given a set of phylogenetic trees with overlapping taxa sets, does there exist a single phylogenetic tree on the union of the taxa sets that contains the information of all input trees?.

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Quaret Compatibility: New Results and Surprising Counterexamples

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  1. Quaret Compatibility: New Results and Surprising Counterexamples Stefan Grünewald

  2. Compatibility Problem Given a set of phylogenetic trees with overlapping taxa sets, does there exist a single phylogenetic tree on the union of the taxa sets that contains the information of all input trees?

  3. Restrictions A restriction of a phylogenetic tree Tto a subset Sof L(T)is the tree obtained from the smallest subtree of Tcontaining Sby suppressing all vertices of degree 2. a b c g e h f d

  4. Restrictions A restriction of a phylogenetic tree Tto a subset S of L(T)is the tree obtained from the smallest subtree of Tcontaining Sby suppressing all vertices of degree 2. c g e h d

  5. Restrictions A restriction of a phylogenetic tree Tto a subset Sof L(T)is the tree obtained from the smallest subtree of Tcontaining Sby suppressing all vertices of degree 2. c g e h d

  6. Displaying T displays a binary treeT’if and T’ is the restriction of T to L(T’). a b c g e h f d

  7. Displaying T displays a binary treeT’if and T’ is the restriction of T to L(T’). Displayed a c a b c g e d h f h d

  8. Displaying T displays a binary treeT’if and T’ is the restriction of T to L(T’). Not displayed a a b b c g e f h f h d

  9. Quartets • A quartet is a binary phylogenetic tree with 4 taxa. • The quartet separating a and b from c and d is denoted by ab|cd. ab|cdac|bdad|bc c b b a a a b c d d d c

  10. Compatibility For a set Pof binary phylogenetic trees, let L(P)be the union of the taxa sets of all trees in P. P is compatible if there is a tree Twith L(T)=L(P)such that Tdisplays every tree in P. If Tis unique then P definesT.

  11. Compatibility • It is NP-complete to decide if a given set of binary phylogenetic trees is compatible, even if all trees are quartets (Steel 1992). • It can be decided in polynomial time if a set Q of quartets with |Q|=|L(Q)|-3defines a phylogenetic tree (Böcker, Dress, Steel 1999). • Given a set Q of quartets and a tree T that displays Q. The complexity of deciding if Q defines T is unknown.

  12. Thin Quartet Sets A set Qof quartets is thin if every subset of the taxa set with k≥4 elements contains at most k-3 quartets. Theorem 1: If Q is thin then Q is compatible.

  13. Maximal Hierarchies Theorem 2 (Böcker, Dress, Steel, 1999): Every minimum defining quartet set contains a maximal hierarchy of excess-free subsets. Dezulian, Steel (2004): “… one of the most mysterious and apparently difficult results in phylogeny.”

  14. Maximal Hierarchies bg|af ah|bf cd|gh cf|dg

  15. Maximal Hierarchies g b a f h bg|af ah|bf cd|gh cf|dg

  16. Maximal Hierarchies g c g b a f h f h d bg|af ah|bf cd|gh cf|dg

  17. Maximal Hierarchies c g b a f h g d c g b a f h f h d bg|af ah|bf cd|gh cf|dg

  18. Freely compatible quadruple sets A set of quadruples is freely compatible if, for every choice of one quartet for each quadruple, the obtained quartet set is compatible. A set of quadruples is thin if every subset of the taxa set with k≥4 elements contains at most k-3 quadruples.

  19. Freely compatible quadruple sets By Theorem 1, every thin set of quadruples is freely compatible. Question: Is every freely compatible set of quadruples thin?

  20. Freely compatible quadruple sets By Theorem 1, every thin set of quadruples is freely compatible. Question: Is every freely compatible set of quadruples thin? Answer: No! A set of 13 quadruples on 13 taxa such that every two quadruples intersect in exactly two taxa is a counterexample (SG, Li Junrui).

  21. The Quartet Graph b c a d f e ab|de, bc|ef, cd|fa

  22. The Quartet Graph b c a d f e ab|de, bc|ef, cd|fa c b d a f e

  23. The Quartet Graph c a,b d f e b ab|de, bc|ef, cd|fa c a d f e

  24. The Quartet Graph c a,b d e,f b ab|de, bc|ef, cd|fa c a d e f

  25. The Quartet Graph c,d a,b e,f c b ab|de, bc|ef, cd|fa d a e f

  26. Closure Rules a b c d e ab|cd, bc|de infer ab|ce, ab|de, ac|de. A quartet set is called closed if no closure rule can be applied to any subset.

  27. An Example The quartet set corresponding to the picture is strongly closed but not compatible (SG, Steel, Swenson, 2007).

  28. An incompatible quartet set with little overlap In1992, Mike Steel used a counting argument to show that are incompatible quartet sets such that two quartets in the set have at most one taxon in common.

  29. An Example

  30. An Example

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