{
  "id": "2207.08981",
  "version": "v1",
  "published": "2022-07-18T23:43:21.000Z",
  "updated": "2022-07-18T23:43:21.000Z",
  "title": "A splitter theorem for elastic elements in $3$-connected matroids",
  "authors": [
    "George Drummond",
    "Charles Semple"
  ],
  "comment": "22 pages, 2 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "An element $e$ of a $3$-connected matroid $M$ is elastic if ${\\rm si}(M/e)$, the simplification of $M/e$, and ${\\rm co}(M\\backslash e)$, the cosimplification of $M\\backslash e$, are both $3$-connected. It was recently shown that if $|E(M)|\\geq 4$, then $M$ has at least four elastic elements provided $M$ has no $4$-element fans and no member of a specific family of $3$-separators. In this paper, we extend this wheels-and-whirls type result to a splitter theorem, where the removal of elements is with respect to elasticity and keeping a specified $3$-connected minor. We also prove that if $M$ has exactly four elastic elements, then it has path-width three. Lastly, we resolve a question of Whittle and Williams, and show that past analogous results, where the removal of elements is relative to a fixed basis, are consequences of this work.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-18T23:43:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B35"
    ],
    "keywords": [
      "elastic elements",
      "splitter theorem",
      "connected matroid",
      "wheels-and-whirls type result",
      "past analogous results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}