{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.13336",
  "author_total": 2,
  "authors": [
    {
      "name": "Anna Lindeberg"
    },
    {
      "name": "Marc Hellmuth"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8ny5tffx19nqq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.13336",
  "fulltext_url": null,
  "id": "8ny5tffx19nqq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.13336",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "q-bio.PE"
    ],
    [
      "openalex",
      "1312|Genomics and Phylogenetic Studies"
    ]
  ],
  "openalex": "W7213270452",
  "pdf_url": "https://arxiv.org/pdf/2609.13336",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8ny5tffx19nqq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.13336",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.13336"
    }
  ],
  "subjects": [
    {
      "field": "biology",
      "origin": "arxiv:q-bio.PE",
      "subfield": "evolutionary-biology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "NOC NOC, who's there? Clustering systems of tree-child and normal networks",
  "title_text": "NOC NOC, who's there? Clustering systems of tree-child and normal networks",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:30Z",
  "url": "https://research.draconitic.com/paper/8ny5tffx19nqq",
  "venue": "arXiv",
  "versions": []
}