{
  "abstract": "We study affine deformations $\\mathcal{A}(G_{\\mathcal{S}})$ of graphic arrangements and their cones. Here $G=([n],E(G))$ is a simple graph, $\\mathcal{S}=(S_{ij})$ is a family of finite gain sets, and $\\mathcal{A}(G_{\\mathcal{S}})$ consists of the hyperplanes $x_i-x_j=a$ with $a\\in S_{ij}$. We call $G_{\\mathcal{S}}$ blockwise admissible if every block has a vertex ordering $v_1,\\ldots,v_m$ satisfying $S_{v_kv_j}-S_{v_kv_i}\\subseteq S_{v_iv_j}$ for every $k$ and all distinct $i,j>k$. Every such ordering is a perfect elimination ordering. We prove, for arbitrary $G$, that the following are equivalent: (i) $G_{\\mathcal{S}}$ is blockwise admissible; (ii) the cone $c\\mathcal{A}(G_{\\mathcal{S}})$ is supersolvable; and (iii) $\\mathcal{A}(G_{\\mathcal{S}})$ admits a nice partition. We give a direct arrangement-theoretic proof of these equivalences. For a block, the equivalence between admissibility and supersolvability is already contained in Zaslavsky's characterization of supersolvable graphic-lift lattices. We further show that every nice partition on a block is induced by an admissible ordering and that its induced edge classes are stars with distinct centers. Under these equivalent conditions, we construct a maximal modular chain through the hyperplane at infinity. We also prove that if $c\\mathcal{A}(G_{\\mathcal{S}})$ is free, then $G$ is chordal.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.39407",
  "author_total": 3,
  "authors": [
    {
      "name": "Weikang Liang"
    },
    {
      "name": "Suijie Wang"
    },
    {
      "name": "Yue Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/01qt6s2bm9emy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.39407",
  "fulltext_url": null,
  "id": "01qt6s2bm9emy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.39407",
  "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": "W7215028335",
  "pdf_url": "https://arxiv.org/pdf/2609.39407",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/01qt6s2bm9emy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.39407",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.39407"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Nice Partitions, Supersolvability, and Freeness in Deformations of Graphic Arrangements",
  "title_text": "Nice Partitions, Supersolvability, and Freeness in Deformations of Graphic Arrangements",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/01qt6s2bm9emy",
  "venue": "arXiv",
  "versions": []
}