{
  "abstract": "Motivated by recent work on tree independence number, we study the path independence number of a graph $G$: the minimum integer $k$ such that there is a path decomposition of $G$ where each bag induces a graph with independence number at most $k$. We show that every graph excluding both an induced forest minor and an induced star has bounded path independence number. This characterises when a graph class that excludes an induced star has bounded path independence number while also partially resolving a conjecture of Dallard, Krnc, Kwon, Milani{č}, Munaro, Štorgel and Wiederrecht (2024). Furthermore, we show that graphs excluding both an apex-forest induced minor and an induced star have bounded tree independence number. As a consequence, for every fixed apex-forest $H$ and integer $t$, there is a polynomial-time algorithm to test whether a $K_{1,t}$-induced-subgraph-free graph contains $H$ as an induced minor. Moreover, it follows that the Maximum Weight Independent Set problem, as well as several other NP-hard problems, can be solved in polynomial-time on $K_{1,t}$-induced-subgraph-free graphs that exclude $H$ as an induced minor.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10406",
  "author_total": 2,
  "authors": [
    {
      "name": "Robert Hickingbotham"
    },
    {
      "name": "Gwenaël Joret"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/84fy886pkn8ft.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10406",
  "fulltext_url": null,
  "id": "84fy886pkn8ft",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.10406",
  "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.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7212107882",
  "pdf_url": "https://arxiv.org/pdf/2609.10406",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/84fy886pkn8ft.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:16Z",
      "native_id": "2609.10406",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10406"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Induced Forest Minor Theorem for Graphs Without an Induced Star",
  "title_text": "Induced Forest Minor Theorem for Graphs Without an Induced Star",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:16Z",
  "url": "https://research.draconitic.com/paper/84fy886pkn8ft",
  "venue": "arXiv",
  "versions": []
}