The $\ominus$-metric to compare phylogenetic networks

2026-07-09

AI Summary

This study addresses the problem of effectively measuring the dissimilarity between rooted phylogenetic networks while preserving ancestral relationships. To this end, the authors propose two novel distance measures based on the graph-theoretic $\ominus$-operator: $d_{\ominus}$, which quantifies network dissimilarity via the minimum number of vertex deletions required to achieve isomorphism, and $d_{\ominus}^-$, which disregards shortcut arcs to focus specifically on ancestral structure comparison. This work introduces, for the first time, vertex-deletion-based network distances that maintain ancestral relations, thereby extending the classical Robinson–Foulds distance. Theoretical analysis shows that $d_{\ominus}^-$ is computable in polynomial time for tree-child, normal, and level-1 networks and admits a 2-approximation algorithm for distinct-cluster networks, whereas $d_{\ominus}$ is NP-hard and not approximable within any constant factor.

Abstract

We introduce two novel distances for comparing rooted phylogenetic networks based on the $\ominus$-operator, which removes a vertex while preserving the ancestor relations among the remaining vertices. The distance $d_{\ominus}$ measures the minimum number of such removals needed to obtain isomorphic networks, whereas $d_{\ominus}^-$ ignores shortcut arcs and therefore compares the induced ancestry structures. We show that $d_{\ominus}$ is a metric up to leaf-fixing isomorphism and that $d_{\ominus}^-$ is a metric up to shortcut-free isomorphism. Moreover, both distances extend the Robinson--Foulds distance on phylogenetic trees and are bounded below by the hardwired cluster distances. For several broad network classes, including tree-child, normal, level-$1$, and regular networks, $d_{\ominus}^-$ can be computed in polynomial time. In contrast, computing $d_{\ominus}$ is NP-hard, W[2]-hard when parameterized by the distance value, and admits no polynomial-time constant-factor approximation unless $\mathrm{P}=\mathrm{NP}$. Although computing $d_{\ominus}^-$ is NP-hard in general, for distinct-cluster networks it reduces to \textsc{Vertex Cover}, yielding a fixed-parameter algorithm and a polynomial-time $2$-approximation.

Problem

• phylogenetic networks
• network comparison
• distance metric
• ancestry structure
• computational complexity

Innovation

• ⊖-metric
• phylogenetic networks
• ancestry-preserving operations
• polynomial-time computability
• NP-hardness