{
  "abstract": "A vertex $v$ of a graph $G$ $ve$-dominates every edge incident with a vertex in its closed neighborhood $N_{G}[v]$. A set of vertices is a vertex-edge dominating set ($ve$-dominating set) if every edge of $G$ is $ve$-dominated by some vertex in the set. The minimum cardinality of such a set is the vertex-edge domination number $γ_{ve}(G)$, also known as the isolation number. In this paper, we study the effect of vertex removal on $γ_{ve}(G)$, establish sharp bounds on $γ_{ve}(G-v)$ and characterize the vertices whose removal decreases, preserves, or increases $γ_{ve}(G)$. Let $V^0(G)$ and $V^+(G)$ denote the sets of vertices whose removal preserves and increases $γ_{ve}(G)$, respectively, and let $V_{\\mathrm{fix}}^0(G)$ consist of the vertices in $V^0(G)$ belonging to every minimum $ve$-dominating set. We prove that \\[ |V^0(G)|\\geq3|V_{\\mathrm{fix}}^0(G)|+\\sum_{v\\in V^+(G)} \\bigl(γ_{ve}(G-v)-γ_{ve}(G)+1\\bigr). \\] In particular, $|V^0(G)|\\geq 2|V^+(G)|$. We also prove that every vertex in $V_{\\mathrm{fix}}^0(G)$ lies on a cycle and that $V^0(T)\\neq\\emptyset$ for every tree $T$ of order at least three.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.26912",
  "author_total": 3,
  "authors": [
    {
      "name": "Minhui Li"
    },
    {
      "name": "Shumin Zhang"
    },
    {
      "name": "Fengming Dong"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3wmrzbw5a1zxy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.26912",
  "fulltext_url": null,
  "id": "3wmrzbw5a1zxy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.26912",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7223150767",
  "pdf_url": "https://arxiv.org/pdf/2610.26912",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3wmrzbw5a1zxy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.26912",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.26912"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Vertex Removal and Vertex-Edge Domination",
  "title_text": "Vertex Removal and Vertex-Edge Domination",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/3wmrzbw5a1zxy",
  "venue": "arXiv",
  "versions": []
}