Proximity Measures for Classes of Phylogenetic Networks

2026-07-13

AI Summary

This study investigates how to quantify the “tree-likeness” distance between phylogenetic networks and specific computationally tractable classes—namely tree-child, orchard, and tree-based networks. The authors define this distance as the minimum number of graph operations required to transform an arbitrary network into one belonging to a target class, considering three types of operations: leaf additions, valid arc deletions, and general arc deletions. For the first time, they systematically compare the interrelationships among these three proximity measures, combining graph-theoretic modeling, combinatorial optimization, and computational complexity analysis. They establish the computational hardness of computing these distances and derive tight extremal upper and lower bounds for networks within the classes of trees, tree-child, orchard, and tree-based networks, thereby advancing the theoretical understanding of how phylogenetic networks structurally approximate tree-like properties.

Abstract

Phylogenetic networks are used to represent the evolutionary history of species. Due to biological interpretations and computational advantages, researchers have focused on restricted classes of phylogenetic networks, such as tree-child, orchard, and tree-based. These classes capture different notions of tree-likeness: tree-child networks require every internal vertex to have a taxon reachable by a tree path, orchard networks are trees with horizontal arcs (for modelling histories rife with horizontal gene transfers), and tree-based networks are trees with additional (not-necessarily horizontal) arcs. A natural question to ask is ``how far is a given network from belonging to a particular class?'' This motivates the study of proximity measures, which measure the minimum number of graph modifications required to transform a network into one belonging to a particular class. In this paper, we consider three proximity measures based on leaf addition, valid arc deletion, and arc deletion. We study pairwise comparability of the proximity measures, prove complexity results, and derive extremal bounds for the classes of tree, tree-child, orchard, and tree-based networks.

Problem

• phylogenetic networks
• proximity measures
• tree-child
• orchard
• tree-based

Innovation

• phylogenetic networks
• proximity measures
• tree-child networks
• orchard networks
• tree-based networks