{
  "id": "1705.03418",
  "version": "v1",
  "published": "2017-05-09T16:38:57.000Z",
  "updated": "2017-05-09T16:38:57.000Z",
  "title": "A notion of minor-based matroid connectivity",
  "authors": [
    "Zachary Gershkoff",
    "James Oxley"
  ],
  "comment": "13 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "For a matroid $N$, a matroid $M$ is $N$-connected if every two elements of $M$ are in an $N$-minor together. Thus a matroid is connected if and only if it is $U_{1,2}$-connected. This paper proves that $U_{1,2}$ is the only connected matroid $N$ such that if $M$ is $N$-connected with $|E(M)| > |E(N)|$, then $M \\backslash e$ or $M / e$ is $N$-connected for all elements $e$. Moreover, we show that $U_{1,2}$ and $M(\\mathcal{W}_2)$ are the only connected matroids $N$ such that, whenever a matroid has an $N$-minor using $\\{e,f\\}$ and an $N$-minor using $\\{f,g\\}$, it also has an $N$-minor using $\\{e,g\\}$. Finally, we show that $M$ is $U_{0,1} \\oplus U_{1,1}$-connected if and only if every clonal class of $M$ is trivial.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-09T16:38:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B35"
    ],
    "keywords": [
      "minor-based matroid connectivity",
      "connected matroid",
      "clonal class"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}