Section: Mathematical & Computational Biology
Topic: Biophysics and computational biology, Computer sciences, Applied mathematics

When Three Trees Go to War

van Iersel, Leo1  ; Jones, Mark1  ; Weller, Mathias2, 3   
1 Delft Institute of Applied Mathematics, Mekelweg 4, 2628 CD, Delft // POSTBUS 5031, 2600 GA Delft
2 Technical University of Berlin / Technische Universität Berlin, Straße des 17. Juni 135 10623 Berlin
3 Institut des sciences informatiques et de leurs interactions - CNRS Sciences informatiques, Centre National de la Recherche Scientifique, 3 Rue Michel Ange 75794 Paris Cedex 16 - France

10.24072/pcjournal.419 - Peer Community Journal, Volume 4 (2024), article no. e54.

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How many reticulations are needed for a phylogenetic network to display a given set of k phylogenetic trees on n leaves? For k = 2, Baroni et al. [Ann. Comb. 8, 391-408 (2005)] showed that the answer is n − 2. Here, we show that, for k ≥ 3 the answer is at least (3 /2 − ε)n. Concretely, we prove that, for each ε > 0, there is some n ∈ N such that three n-leaf caterpillar trees can be constructed in such a way that any network displaying these caterpillars contains at least (3 /2 − ε)n reticulations. The case of three trees is interesting since it is the easiest case that cannot be equivalently formulated in terms of agreement forests. Instead, we base the result on a surprising lower bound for multilabelled trees (MUL-trees) displaying the caterpillars. Indeed, we show that one cannot do (more than an ε) better than the trivial MUL-tree resulting from a simple concatenation of the given caterpillars. The results are relevant for the development of methods for the Hybridization Number problem on more than two trees. This fundamental problem asks to construct a binary phylogenetic network with a minimum number of reticulations displaying a given set of phylogenetic trees.

Published online:
DOI: 10.24072/pcjournal.419
Type: Research article

van Iersel, Leo 1; Jones, Mark 1; Weller, Mathias 2, 3

1 Delft Institute of Applied Mathematics, Mekelweg 4, 2628 CD, Delft // POSTBUS 5031, 2600 GA Delft
2 Technical University of Berlin / Technische Universität Berlin, Straße des 17. Juni 135 10623 Berlin
3 Institut des sciences informatiques et de leurs interactions - CNRS Sciences informatiques, Centre National de la Recherche Scientifique, 3 Rue Michel Ange 75794 Paris Cedex 16 - France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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van Iersel, Leo; Jones, Mark; Weller, Mathias. When Three Trees Go to War. Peer Community Journal, Volume 4 (2024), article  no. e54. doi : 10.24072/pcjournal.419. https://peercommunityjournal.org/articles/10.24072/pcjournal.419/

PCI peer reviews and recommendation, and links to data, scripts, code and supplementary information: 10.24072/pci.mcb.100187

Conflict of interest of the recommender and peer reviewers:
The recommender in charge of the evaluation of the article and the reviewers declared that they have no conflict of interest (as defined in the code of conduct of PCI) with the authors or with the content of the article.

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