{
  "abstract": "We give an $O_H(n^5)$-time algorithm to test whether an $n$-vertex labeled tour graph has a full immersion isomorphic to a labeled tour graph $H$. A labeled tour graph is a connected $4$-regular graph whose edges are labeled by binary vectors of fixed length. To obtain a full immersion, a vertex may be removed only by pairing up its four incident edges and \"splitting off\" both pairs, summing the labels within each pair. Even for one-bit labels, no polynomial-time algorithm was previously known for arbitrary fixed $H$. Our main tool is of independent interest: we give an algorithmic min-max theorem that, in particular, characterizes the maximum number of odd-length circuits in a rooted circuit decomposition of an Eulerian graph. Our results have implications for vertex-minors, whose study is motivated by applications in quantum networking, connections with graph minor theory, and the goal of developing an irrelevant vertex technique for dense graphs. For vertex-minors, circle graphs play the same role as planar graphs do for graph minors. As a corollary of our main theorem, we obtain a fixed-parameter tractable algorithm for apex-vertex-minor testing in $k$-apex circle graphs. We also prove an irrelevant vertex theorem for such graphs of large rank-width. More broadly, our results are essential to forthcoming work of three of the authors proving a structure theorem for graphs with a forbidden vertex-minor. There, our min-max theorem plays the role of both the algorithmic $A$-paths and Two Paths Theorems.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.06776",
  "author_total": 5,
  "authors": [
    {
      "name": "Hadas Barabash"
    },
    {
      "name": "Jim Geelen"
    },
    {
      "name": "Rose McCarty"
    },
    {
      "name": "Caleb McFarland"
    },
    {
      "name": "Paul Wollan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fa0ysz4yew3hx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.06776",
  "fulltext_url": null,
  "id": "fa0ysz4yew3hx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.06776",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.06776",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fa0ysz4yew3hx.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2611.06776",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.06776"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Finding full immersions is fixed-parameter tractable in labeled tour graphs",
  "title_text": "Finding full immersions is fixed-parameter tractable in labeled tour graphs",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:30Z",
  "url": "https://research.draconitic.com/paper/fa0ysz4yew3hx",
  "venue": "arXiv",
  "versions": []
}