{
  "abstract": "For a connected graph $G$ with at least one edge, the \\textit{eternal connected vertex cover} number $ecvc(G)$ is the minimum number of guards that can maintain a connected vertex cover after every response to an arbitrary sequence of edge attacks. Denote the minimum size of a connected vertex cover by $cvc(G)$. It is known that $cvc(G)\\leq ecvc(G)\\leq cvc(G)+1$. A necessary condition for $ecvc(G)=cvc(G)$ is that every vertex belongs to some minimum connected vertex cover. We show that this condition is not sufficient: there exists a $32$-vertex graph $G$ that has $cvc(G)=19$ and $ecvc(G)=20$, although every vertex of $G$ belongs to some minimum connected vertex cover. For connected graphs with minimum degree at least two, we establish the sharp bound $cvc(G)\\geq 2|V(G)|-|E(G)|-1$ and $\\mathcal F$ denotes the class attaining equality. We prove that $G\\in\\mathcal F$ if and only if the vertices of degree at least three induce a forest. Within $\\mathcal F$, the conditions $ecvc(G)=cvc(G)$, membership of every vertex in some minimum connected vertex cover, and the presence of at least two degree-two vertices on every cycle are equivalent. As applications, we obtain $ecvc(G)=cvc(G)$ for full subdivisions of connected graphs of minimum degree at least two and for minimally $2$-connected graphs. In both the families, every minimum connected vertex cover is an eternally winning configuration. This is not true in general for graphs outside $\\mathcal{F}$, we show one example of such a graph.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.39103",
  "author_total": 2,
  "authors": [
    {
      "name": "Rajat Adak"
    },
    {
      "name": "Saraswati Girish Nanoti"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d5mag34pgbsqq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.39103",
  "fulltext_url": null,
  "id": "d5mag34pgbsqq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.39103",
  "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",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7215094716",
  "pdf_url": "https://arxiv.org/pdf/2609.39103",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d5mag34pgbsqq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.39103",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.39103"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On Eternal Connected Vertex Cover",
  "title_text": "On Eternal Connected Vertex Cover",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/d5mag34pgbsqq",
  "venue": "arXiv",
  "versions": []
}