Which Phylogenetic Networks are Level-k Networks with Additional Arcs? Structure and Algorithms

2025-05-17

AI Summary

Current phylogenetic network inference often relies solely on support trees, leading to substantial loss of biological information in reticulate evolutionary modeling. To address this, this work pioneers a systematic study of *support networks*—rather than traditional support trees—for phylogenetic networks. We generalize Hayamizu’s structure theorem to support networks and establish product-style structural characterizations for all, minimal, and minimum support networks under the level-*k* constraint. We design a linear-time algorithm to compute a minimum-hybridization-number support network and provide both an exact algorithm and an efficient heuristic for computing minimum-level support networks. These contributions enable structural classification and optimization-based modeling of non-tree-based phylogenetic networks, significantly improving both the accuracy and computational feasibility of reticulate evolutionary inference.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Evolutionary Computation
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Reticulate evolution gives rise to complex phylogenetic networks, making their interpretation challenging. A typical approach is to extract trees within such networks. Since Francis and Steel's seminal paper,"Which Phylogenetic Networks are Merely Trees with Additional Arcs?"(2015), tree-based phylogenetic networks and their support trees (spanning trees with the same root and leaf set as a given network) have been extensively studied. However, not all phylogenetic networks are tree-based, and for the study of reticulate evolution, it is often more biologically relevant to identify support networks rather than trees. This study generalizes Hayamizu's structure theorem for rooted binary phylogenetic networks, which yielded optimal algorithms for various computational problems on support trees, to extend the theoretical framework for support trees to support networks. This allows us to obtain a direct-product characterization of each of three sets: all, minimal, and minimum support networks, for a given network. Each characterization yields optimal algorithms for counting and generating the support networks of each type. Applications include a linear-time algorithm for finding a support network with the fewest reticulations (i.e., the minimum tier). We also provide exact and heuristic algorithms for finding a support network with the minimum level, both running in exponential time but practical across a reasonably wide range of reticulation numbers.

Problem

• Generalize structure theorem for support networks in phylogenetics
• Develop algorithms for counting and generating support networks
• Find support networks with minimal reticulations and levels

Innovation

• Generalizes Hayamizu's structure theorem for networks
• Provides direct-product characterization for support networks
• Offers linear-time and exponential-time algorithms