# Rearrangement operations on unrooted phylogenetic networks

## TL;DR

This paper explores the properties of spaces of unrooted phylogenetic networks under rearrangement operations like NNI, SPR, and TBR, including connectivity, diameter bounds, and computational complexity of distance measures.

## Contribution

It extends known rearrangement operations from trees to networks, analyzing their properties and computational complexity in this broader context.

## Key findings

- Proved connectedness of network spaces under these operations
- Established asymptotic bounds on the diameters of network spaces
- Showed computing TBR and PR distances is NP-hard

## Abstract

Rearrangement operations transform a phylogenetic tree into another one and hence induce a metric on the space of phylogenetic trees. Popular operations for unrooted phylogenetic trees are NNI (nearest neighbour interchange), SPR (subtree prune and regraft), and TBR (tree bisection and reconnection). Recently, these operations have been extended to unrooted phylogenetic networks, which are generalisations of phylogenetic trees that can model reticulated evolutionary relationships. Here, we study global and local properties of spaces of phylogenetic networks under these three operations. In particular, we prove connectedness and asymptotic bounds on the diameters of spaces of different classes of phylogenetic networks, including tree-based and level-k networks. We also examine the behaviour of shortest TBR-sequence between two phylogenetic networks in a class, and whether the TBR-distance changes if intermediate networks from other classes are allowed: for example, the space of phylogenetic trees is an isometric subgraph of the space of phylogenetic networks under TBR. Lastly, we show that computing the TBR-distance and the PR-distance of two phylogenetic networks is NP-hard.