{
  "id": "2409.13834",
  "version": "v1",
  "published": "2024-09-20T18:20:24.000Z",
  "updated": "2024-09-20T18:20:24.000Z",
  "title": "Detachable pairs in $3$-connected matroids and simple $3$-connected graphs",
  "authors": [
    "Nick Brettell",
    "Charles Semple",
    "Gerry Toft"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $M$ be a $3$-connected matroid. A pair $\\{e,f\\}$ in $M$ is detachable if $M \\backslash e \\backslash f$ or $M / e / f$ is $3$-connected. Williams (2015) proved that if $M$ has at least 13 elements, then at least one of the following holds: $M$ has a detachable pair, $M$ has a $3$-element circuit or cocircuit, or $M$ is a spike. We address the case where $M$ has a $3$-element circuit or cocircuit, to obtain a characterisation of when a matroid with at least 13 elements has a detachable pair. As a consequence, we characterise when a simple $3$-connected graph $G$ with $|E(G)| \\ge 13$ has a pair of edges $\\{e,f\\}$ such that $G/e/f$ or $G \\backslash e\\backslash f$ is simple and $3$-connected.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-20T18:20:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B35"
    ],
    "keywords": [
      "detachable pair",
      "connected matroid",
      "connected graph",
      "element circuit",
      "characterisation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}