A Class of Unrooted Phylogenetic Networks Inspired by the Properties of Rooted Tree-Child Networks

2026-03-07

AI Summary

This work proposes the class of q-separable networks to identify structures in unrooted phylogenetic networks that serve as analogues to rooted tree-child subnetworks, balancing expressiveness and computational tractability. The class is formally defined for the first time and exhibits favorable algorithmic properties: for any fixed \( q \geq 1 \), membership can be decided in polynomial time, and when \( q \geq 3 \), the otherwise NP-hard tree containment problem becomes polynomial-time solvable. Through graph-theoretic and computational complexity analyses, the study elucidates the relationship between network orientability and structural constraints, further proving that recognizing general tree-child orientable networks is NP-hard—thereby highlighting the theoretical and algorithmic advantages of q-separable networks.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Social Networks and Social Media: Social media analysis through the lenses of networks
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

A directed phylogenetic network is tree-child if every non-leaf vertex has a child that is not a reticulation. As a class of directed phylogenetic networks, tree-child networks are very useful from a computational perspective. For example, several computationally difficult problems in phylogenetics become tractable when restricted to tree-child networks. At the same time, the class itself is rich enough to contain quite complex networks. Furthermore, checking whether a directed network is tree-child can be done in polynomial time. In this paper, we seek a class of undirected phylogenetic networks that is rich and computationally useful in a similar way to the class tree-child directed networks. A natural class to consider for this role is the class of tree-child-orientable networks which contains all those undirected phylogenetic networks whose edges can be oriented to create a tree-child network. However, we show here that recognizing such networks is NP-hard, even for binary networks, and as such this class is inappropriate for this role. Towards finding a class of undirected networks that fills a similar role to directed tree-child networks, we propose new classes called $q$-cuttable networks, for any integer $q\geq 1$. We show that these classes have many of the desirable properties, similar to tree-child networks in the rooted case, including being recognizable in polynomial time, for all $q\geq 1$. Towards showing the computational usefulness of the class, we show that the NP-hard problem Tree Containment is polynomial-time solvable when restricted to $q$-cuttable networks with $q\geq 3$.

Problem

• phylogenetic networks
• tree-child networks
• undirected networks
• computational tractability
• network orientation

Innovation

• phylogenetic networks
• tree-child networks
• q-cuttable networks
• computational complexity
• Tree Containment