{
  "abstract": "Let $(G_n)_{n\\ge n_0}$ be a family of graphs on $[n]$ whose edge ideals $I_n\\subseteq R_n=k[x_1,\\dots,x_n]$ form an $\\mathrm{Inc}(\\mathbb{N})$-invariant chain, $I_{n+r}=\\mathrm{Inc}(\\mathbb{N})_{n,n+r}(I_n)$ for $n\\ge n_0,\\ r\\ge0$. We determine which pairs $(n,r)$ make $R_{n+r}/I_{n+r}$ Cohen--Macaulay, for five classical families: line graphs $L_n$, complements of line graphs $L_n^c$, complete graphs $K_n$, cyclic graphs $C_n$, and complements of cyclic graphs $C_n^c$. For line graphs we give the generators of $I_{n+r}$, the height and Krull dimension of $R_{n+r}/I_{n+r}$, and a complete classification: for $n\\ge5$, $R_{n+r}/I_{n+r}$ is Cohen--Macaulay if and only if $r=n-4$. For complements of line graphs, complete graphs, and complements of cyclic graphs, we show the chain is Cohen--Macaulay unconditionally, for every $r\\ge0$; the first and third arise from the same underlying phenomenon, in which the $\\mathrm{Inc}(\\mathbb{N})$-invariant chain reproduces the original graph itself at each step. For cyclic graphs we prove $\\mathrm{Inc}(\\mathbb{N})_{n,n+r}(I(C_n))=I(K_{n+r})$ once $r\\ge n-3$, giving Cohen--Macaulayness in this range, and conjecture -- with supporting computational and structural evidence -- a complete classification: Cohen--Macaulayness holds if and only if $r\\ge\\lfloor(n-4)/2\\rfloor$ and $r\\ne n-4$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03566",
  "author_total": 3,
  "authors": [
    {
      "name": "Imran Anwar"
    },
    {
      "name": "Mughees Ghayas"
    },
    {
      "name": "Azhar Javed"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/99ftjdq45afth.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03566",
  "fulltext_url": null,
  "id": "99ftjdq45afth",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03566",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AC"
    ],
    [
      "openalex",
      "2602|Commutative Algebra and Its Applications"
    ]
  ],
  "openalex": "W7208727863",
  "pdf_url": "https://arxiv.org/pdf/2609.03566",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/99ftjdq45afth.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2609.03566",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03566"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Cohen--Macaulayness of $\\mathrm{Inc}(\\mathbb{N})$-Invariant Chains of Edge Ideals",
  "title_text": "Cohen--Macaulayness of Inc(ℕ)-Invariant Chains of Edge Ideals",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/99ftjdq45afth",
  "venue": "arXiv",
  "versions": []
}