### Abstract

Compatibility of unrooted phylogenetic trees is a well studied problem in phylogenetics. It asks to determine whether for a set of k input trees T 1,...,T k there exists a larger tree (called a supertree) that contains the topologies of all k input trees. When any such supertree exists we call the instance compatible and otherwise incompatible. It is known that the problem is NP-hard and FPT, although a constructive FPT algorithm is not known. It has been shown that whenever the treewidth of an auxiliary structure known as the display graph is strictly larger than the number of input trees, the instance is incompatible. Here we show that whenever the treewidth of the display graph is at most 2, the instance is compatible. Furthermore, we give a polynomial-time algorithm to construct a supertree in this case. Finally, we demonstrate both compatible and incompatible instances that have display graphs with treewidth 3, highlighting that the treewidth of the display graph is (on its own) not sufficient to determine compatibility.

Original language | English |
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Title of host publication | Algorithms for Computational Biology |

Subtitle of host publication | AICoB 2014 International Conference on Algorithms for Computational Biology |

Editors | A. Horia Dediu, C. Martín-Vide, B. Truthe |

Place of Publication | Switzerland |

Publisher | Springer |

Pages | 71-82 |

Number of pages | 12 |

ISBN (Print) | 978-3-319-07952-3 |

DOIs | |

Publication status | Published - 1 Jan 2014 |

### Publication series

Series | Lecture Notes in Computer Science |
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Number | 8542 |

## Cite this

Grigoriev, A., Kelk, S., & Lekic, L. (2014). On low treewidth graphs and supertrees. In A. Horia Dediu, C. Martín-Vide, & B. Truthe (Eds.),

*Algorithms for Computational Biology: AICoB 2014 International Conference on Algorithms for Computational Biology*(pp. 71-82). Springer. Lecture Notes in Computer Science, No. 8542 https://doi.org/10.1007/978-3-319-07953-0_6