{
  "abstract": "The classical cactus theorem of Dinitz, Karzanov and Lomonosov from 1976 states that the minimum edge-cuts of a connected graph are displayed by a cactus. For vertex-separations, Tutte's theorem gives an exact cactus-decomposition for connectivity $2$, but exact cactus-decompositions fail for higher connectivity. Extending earlier approximate cactus-decomposition results for connectivities $3$ and $4$ by Carmesin and Kurkofka [FOCS 2023] and Kurkofka and Planken [SODA 2026] to all connectivities, we prove that for every $k\\in\\mathbb{N}$ there is a constant $d=d(k)$ such that every $k$-connected graph $G$ admits a cactus-decomposition displaying $k$-separations such that every $k$-separation of $G$ has edit-distance at most $d$ to a displayed separation. Our proof develops tools that may be useful beyond this result. The first is a three-dimensional version of the classical corner-diagram method, which we use to show that triples of crossing $k$-separations have a controlled two-dimensional behaviour. The second is a covering-space approach for studying local crossing patterns from a global perspective. Third, we advance the theory of flowers, which originated in matroid theory.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01772",
  "author_total": 4,
  "authors": [
    {
      "name": "Romain Bourneuf"
    },
    {
      "name": "Johannes Carmesin"
    },
    {
      "name": "Jan Kurkofka"
    },
    {
      "name": "Tim Planken"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6yqw6kw4a7hj1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01772",
  "fulltext_url": null,
  "id": "6yqw6kw4a7hj1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01772",
  "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",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.01772",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6yqw6kw4a7hj1.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.01772",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01772"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "An approximate cactus-decomposition for vertex-connectivity",
  "title_text": "An approximate cactus-decomposition for vertex-connectivity",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/6yqw6kw4a7hj1",
  "venue": "arXiv",
  "versions": []
}