{
  "abstract": "A connected graph $G$ is matching covered if it contains at least one edge and every edge belongs to some perfect matching of $G$. A matching covered graph is a brick if it is 3-connected and after removing any two vertices, the resulting graph has a perfect matching. Bricks are the building blocks of matching covered graphs. A brick $G$ is extremal if the incidence vectors of all perfect matchings are linearly independent over $\\mathbb{R}$. Zhou et al. [On planar extremal nonsolid bricks, J. Graph Theory, 2026] proved that for every planar extremal brick $G$, there exists a sequence of planar extremal bricks $G_0,G_1,\\dots,G_n$ such that the underlying graph of $G_n$ is an odd wheel, $G_0=G$, and each $G_i$ $(0\\leq i < n)$ has a triangle $T_i$ such that $G_i/T_i = G_{i+1}$, where $G_i/T_i$ means contracting the triangle $T_i$ into a single vertex. In this paper, the result of Zhou et al. is generalized to planar PM-compact bricks, a larger class of bricks including extremal bricks. We present a chain theorem for planar PM-compact bricks: for every planar PM-compact brick $G$, there exists a sequence of planar PM-compact bricks $G_0,G_1,\\dots,G_n$ such that the underlying graph of $G_n$ is an odd wheel, $G_0=G$, and each $G_i$ $(0\\leq i < n)$ contains a triangle $T_i$ such that $G_i/T_i = G_{i+1}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15534",
  "author_total": 3,
  "authors": [
    {
      "name": "Yaxian zhang"
    },
    {
      "name": "Qiuli Li"
    },
    {
      "name": "Fuliang Lu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8csyfxwa50264.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15534",
  "fulltext_url": null,
  "id": "8csyfxwa50264",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15534",
  "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",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7221829590",
  "pdf_url": "https://arxiv.org/pdf/2610.15534",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8csyfxwa50264.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.15534",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15534"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A chain theorem for planar PM-compact bricks",
  "title_text": "A chain theorem for planar PM-compact bricks",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/8csyfxwa50264",
  "venue": "arXiv",
  "versions": []
}