{
  "abstract": "Phylogenetic networks generalize phylogenetic trees as models of evolutionary history, allowing lineages to merge as well as to diverge. For many types of genetic data the root position of such a network cannot be recovered, so that only a semi-directed network can be inferred: a mixed graph in which only the edges entering a reticulation vertex are directed. A common strategy for inferring such a network is to first infer the subnetwork it induces on each set of $k\\geq 3$ of its leaves, called a $k$-net, and then to assemble these pieces. This can only succeed if the $k$-nets determine the network, in which case that network is said to be encoded by its $k$-nets. Semi-directed networks of level-1 and 2, those whose biconnected components contain at most one, respectively two, reticulations, are known to be encoded by their $4$-nets, or quarnets, whereas level-3 networks are not. Even so, in this paper we show that level-3 semi-directed networks are encoded by their $5$-nets, or quinnets, and we characterize the limitation of quarnets exactly: we show that a single previously reported counterexample captures the only obstruction, every other level-3 network being encoded by its quarnets. Our proofs rest on a collection of encoding results for individual structural features of a network, which we establish for networks of arbitrary level and which are of independent interest.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.07855",
  "author_total": 4,
  "authors": [
    {
      "name": "Niels Holtgrefe"
    },
    {
      "name": "Katharina T. Huber"
    },
    {
      "name": "Leo van Iersel"
    },
    {
      "name": "Vincent Moulton"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cxambj5pr6pm8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.07855",
  "fulltext_url": null,
  "id": "cxambj5pr6pm8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.07855",
  "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",
      "math.CO"
    ],
    [
      "arxiv",
      "q-bio.PE"
    ],
    [
      "openalex",
      "1312|Genomics and Phylogenetic Studies"
    ]
  ],
  "openalex": "W7221037038",
  "pdf_url": "https://arxiv.org/pdf/2610.07855",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cxambj5pr6pm8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.07855",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.07855"
    }
  ],
  "subjects": [
    {
      "field": "biology",
      "origin": "arxiv:q-bio.PE",
      "subfield": "evolutionary-biology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Encoding level-3 semi-directed phylogenetic networks by quarnets and quinnets",
  "title_text": "Encoding level-3 semi-directed phylogenetic networks by quarnets and quinnets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:27Z",
  "url": "https://research.draconitic.com/paper/cxambj5pr6pm8",
  "venue": "arXiv",
  "versions": []
}