{
  "abstract": "Let $G$ be a simple graph with cover ideal $J(G)$ in a polynomial ring $S = \\k[x_1, \\ldots, x_n]$. For each integer $t \\ge 1$, let $α_t(G)$ denote the maximum size of an ordered matching $M$ of $G$ with alternating path length $\\ell(M) \\le 2t - 1$. Hang, Tam, and Vu established that \\[ \\operatorname{depth}\\!\\left(S/J(G)^{(t)}\\right) \\le n - 1 - α_t(G) \\] for all $t \\ge 1$, where $J(G)^{(t)}$ is the $t$-th symbolic power of $J(G)$. In this paper, we introduce a natural hereditary class of graphs $\\mathcal{P}_\\k$ for which this inequality holds with equality for every $t \\ge 1$. We prove that $\\mathcal{P}_\\k$ contains all distance-hereditary graphs, and we show that within the class of chordal graphs, $\\mathcal{P}_\\k$ coincides with the class of distance-hereditary (or gem-free) graphs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.09316",
  "author_total": 3,
  "authors": [
    {
      "name": "Nguyen Thu Hang"
    },
    {
      "name": "Nguyen Thi Thanh Tam"
    },
    {
      "name": "Thanh Vu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fd9srcegq5a0g.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.09316",
  "fulltext_url": null,
  "id": "fd9srcegq5a0g",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.09316",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AC"
    ],
    [
      "openalex",
      "2602|Commutative Algebra and Its Applications"
    ]
  ],
  "openalex": "W7221017143",
  "pdf_url": "https://arxiv.org/pdf/2610.09316",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fd9srcegq5a0g.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2610.09316",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.09316"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Structural properties of admissible subgraphs and the depth of symbolic powers of cover ideals",
  "title_text": "Structural properties of admissible subgraphs and the depth of symbolic powers of cover ideals",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/fd9srcegq5a0g",
  "venue": "arXiv",
  "versions": []
}