# How tree-based is my network? Proximity measures for unrooted phylogenetic networks

## TL;DR

This paper introduces measures to quantify how close unrooted phylogenetic networks are to being tree-based, providing new characterizations and a subclassification called tree-based rank, with implications for computational complexity.

## Contribution

It formalizes proximity measures for unrooted phylogenetic networks, defines tree-based rank, and relates unrooted and rooted networks, clarifying decision problem complexities.

## Key findings

- Proximity measures effectively quantify closeness to tree-based networks.
- Tree-based rank identifies networks that are 'very' tree-based.
- Decision problems are polynomial for rooted but NP-complete for unrooted networks.

## Abstract

Tree-based networks are a class of phylogenetic networks that attempt to formally capture what is meant by "tree-like" evolution. A given non-tree-based phylogenetic network, however, might appear to be very close to being tree-based, or very far. In this paper, we formalise the notion of proximity to tree-based for unrooted phylogenetic networks, with a range of proximity measures. These measures also provide characterisations of tree-based networks. One measure in particular, related to the nearest neighbour interchange operation, allows us to define the notion of "tree-based rank". This provides a subclassification within the tree-based networks themselves, identifying those networks that are "very" tree-based. Finally, we prove results relating tree-based networks in the settings of rooted and unrooted phylogenetic networks, showing effectively that an unrooted network is tree-based if and only if it can be made a rooted tree-based network by rooting it and orienting the edges appropriately. This leads to a clarification of the contrasting decision problems for tree-based networks, which are polynomial in the rooted case but NP complete in the unrooted.