{
  "abstract": "The cube polynomial $C_G(x)$ generates the number of $k$-cubes on a graph $G$. As a subclass of partial cubes, the tope graphs of lopsided sets (LOPs) generalize daisy cubes and median graphs. In this paper, we prove that every tope graph of a LOP shares its cube polynomial with some daisy cube, thereby answering affirmatively a problem posed earlier by the authors. Furthermore, we present explicit expressions for the cube polynomials of tope graphs of LOPs: $C_G(x)=f_\\mathcal{K}(x+1)$, where $f_\\mathcal{K}(x)$ is the $f$-polynomial of the cubical complex $\\mathcal{K}$ of $G$. Finally, we show that a class $\\mathcal{G}$ of partial cubes is the class of tope graphs of LOPs if and only if the following equivalent conditions hold: (a) $\\mathcal{G}$ is the maximal pc-minor-closed class such that every graph in $\\mathcal{G}$ has the same cube polynomial as some daisy cube; (b) for every $G\\in\\mathcal{G}$, each antipodal subgraph of $G$ has the same cube polynomial as some daisy cube.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22741",
  "author_total": 3,
  "authors": [
    {
      "name": "Xuan Zheng"
    },
    {
      "name": "Yan-Ting Xie"
    },
    {
      "name": "Shou-Jun Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/74n90f43z1nzp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22741",
  "fulltext_url": null,
  "id": "74n90f43z1nzp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22741",
  "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",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7214031690",
  "pdf_url": "https://arxiv.org/pdf/2609.22741",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/74n90f43z1nzp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.22741",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22741"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The polynomial characterization of tope graphs of the lopsided sets",
  "title_text": "The polynomial characterization of tope graphs of the lopsided sets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/74n90f43z1nzp",
  "venue": "arXiv",
  "versions": []
}