### Abstract

Here we show that deciding whether two rooted binary phylogenetic trees on the same set of taxa permit a cherry-picking sequence, a special type of elimination order on the taxa, is NP-complete. This improves on an earlier result which proved hardness for eight or more trees. Via a known equivalence between cherry-picking sequences and temporal phylogenetic networks, our result proves that it is NP-complete to determine the existence of a temporal phylogenetic network that contains topological embeddings of both trees. The hardness result also greatly strengthens previous inapproximability results for the minimum temporal-hybridization number problem. This is the optimization version of the problem where we wish to construct a temporal phylogenetic network that topologically embeds two given rooted binary phylogenetic trees and that has a minimum number of indegree-2 nodes, which represent events such as hybridization and horizontal gene transfer. We end on a positive note, pointing out that fixed parameter tractability results in this area are likely to ensure the continued relevance of the temporal phylogenetic network model.

Original language | English |
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Pages (from-to) | 131-143 |

Number of pages | 13 |

Journal | Discrete Applied Mathematics |

Volume | 260 |

Early online date | 2019 |

DOIs | |

Publication status | Published - 15 May 2019 |

### Keywords

- COMPLEXITY
- Elimination orders
- NP-hardness
- PHYLOGENETIC NETWORKS
- Phylogenetic networks
- Phylogenetics
- Satisfiability
- TEMPORAL HYBRIDIZATION NUMBER
- Temporal networks

## Cite this

Döcker, J., Van Iersel, L., Kelk, S., & Linz, S. (2019). Deciding the existence of a cherry-picking sequence is hard on two trees.

*Discrete Applied Mathematics*,*260*, 131-143. https://doi.org/10.1016/j.dam.2019.01.031