{
  "abstract": "Motivated by Mader's conjecture on connectivity keeping trees, we study trees obtained from paths by adding one pendant vertex, as well as related problems in triangle-free graphs. For an integer $m$ and $1\\leq i\\leq m-1$, let $P_m^+(i)$ denote the tree obtained from a path of order $m-1$ by adding one pendant vertex adjacent to its $i$th vertex. We prove that, for positive integers $k,m,1\\leq i\\leq m-1$, every $k$-connected graph $G$ with $δ(G)\\geq \\lfloor \\frac{3k}{2}\\rfloor+m-1$ contains a subgraph $T\\cong P_m^+(i)$ such that $κ(G-V(T))\\geq k$. This confirms Mader's conjecture for all pendant extensions of paths. For highly connected triangle-free graphs, a connectivity keeping result for paths was obtained in [J. Combin. Theory Ser. B, 174 (2025), 190-206]. Let $(X,Y)$ be the bipartition of $P_m^+(i)$. We further prove that every $k$-connected triangle-free graph $G$ with $δ(G)\\geq k+\\max\\{|X|,|Y|\\}+[P_m^+(i)\\text{ is bad}]$ contains a subgraph $T\\cong P_m^+(i)$ such that $κ(G-V(T))\\geq k$, where we use Iverson's convention for $[P_m^+(i)\\text{ is bad}]$. This extends the corresponding result for paths to pendant extensions of paths.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23634",
  "author_total": 4,
  "authors": [
    {
      "name": "Menghan Ma"
    },
    {
      "name": "Qinghai Liu"
    },
    {
      "name": "Liping Zhang"
    },
    {
      "name": "Yanmei Hong"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/77y27r5qkfhr8.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23634",
  "fulltext_url": null,
  "id": "77y27r5qkfhr8",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23634",
  "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": "W7214002569",
  "pdf_url": "https://arxiv.org/pdf/2609.23634",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/77y27r5qkfhr8.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.23634",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23634"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Connectivity keeping pendant extensions of paths in $k$-connected graphs and triangle-free graphs",
  "title_text": "Connectivity keeping pendant extensions of paths in k-connected graphs and triangle-free graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/77y27r5qkfhr8",
  "venue": "arXiv",
  "versions": []
}