{
  "abstract": "We show that Max Independent Set (MIS) remains NP-hard in graphs excluding the $5 \\times 5$ grid as an induced minor. This negatively answers a question of Dallard--Milanič--Štorgel that has attracted sustained attention, and refutes a conjecture by Gartland and Lokshtanov. Unless $N$-variable 3-SAT can be solved in time $2^{\\mathcal O(N^{1-\\varepsilon})}$ for some $\\varepsilon > 0$, MIS requires $2^{Ω(n^{1/2-o(1)})}$ time on $n$-vertex graphs in this class. This complements a $2^{\\widetilde{O}(n^{2/3})}$-time algorithm on any $H$-induced-minor-free class by Korhonen and Lokshtanov. Exploiting a connection between low twin-width or reduced parameters on one hand and planar-induced-minor-freeness on the other, we further show that 3-Coloring, Feedback Vertex Set and Dominating Set are NP-complete on weakly sparse graphs of reduced component max-leaf (hence twin-width) at most 3, even when a witnessing contraction sequence is given. Classes of bounded reduced component max-leaf (by any constant) and of twin-width at most 3 are known to exclude a fixed planar graph as an induced minor. The 3-Coloring and Feedback Vertex Set results refute more cases of the Gartland--Lokshtanov conjecture, even when the inputs are weakly sparse. The complexity of MIS in weakly sparse graphs excluding a planar induced minor remains open.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11285",
  "author_total": 2,
  "authors": [
    {
      "name": "Édouard Bonnet"
    },
    {
      "name": "Yeonsu Chang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3h69fk6cavp5g.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11285",
  "fulltext_url": null,
  "id": "3h69fk6cavp5g",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11285",
  "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.CC"
    ],
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      "arxiv",
      "cs.DM"
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      "arxiv",
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    [
      "openalex",
      "1703|Advanced Graph Theory Research"
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  "openalex": "W7212218725",
  "pdf_url": "https://arxiv.org/pdf/2609.11285",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3h69fk6cavp5g.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.11285",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11285"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Max Independent Set Remains NP-hard when Excluding a Planar Induced Minor",
  "title_text": "Max Independent Set Remains NP-hard when Excluding a Planar Induced Minor",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/3h69fk6cavp5g",
  "venue": "arXiv",
  "versions": []
}