NOC NOC, who's there? Clustering systems of tree-child and normal networks

2026-09-11

AI Summary

研究通过NOC属性解决树子和正常网络的聚类系统问题，提供了一种新的等价条件，并由此得到多项式时间识别算法。

Abstract

Clustering systems provide a natural way to encode structural information contained in phylogenetic networks. In this note, we study the clustering systems of normal and tree-child networks through an overlap-based property of set systems, called not-overlap-covered (NOC). We show that the NOC property is equivalent to inclusion-visibility, a memberwise formulation of the strict-compatibility condition previously used for tree-child clustering systems. We characterize normal networks as precisely the semi-regular networks whose clustering systems satisfy NOC. Consequently, a clustering system is realized by a normal network if and only if it satisfies NOC, or equivalently, if every one of its clusters is inclusion-visible. In this case, the Hasse diagram provides a canonical normal realization. These are exactly the clustering systems realized by tree-child networks. The NOC formulation yields a sharp quadratic upper bound on the number of distinct clusters of tree-child and normal networks and a direct polynomial-time recognition algorithm. Finally, we explore several consequences of the NOC perspective beyond the phylogenetic setting. These include connections to the enumeration of normal networks, an order-theoretic interpretation of inclusion-visibility, structural properties of NOC set systems, and a tractable special case of Minimum Set Cover, which is NP-hard in general.

Problem

• Clustering systems
• Phylogenetic networks
• Non-overlap-covered (NOC)
• Tree-child networks
• Normal networks

Innovation

• NOC (Not-Overlap-Covered)
• Inclusion-Visibility
• Normal Networks
• Tree-Child Networks
• Clustering Systems