{
  "id": "2309.13138",
  "version": "v1",
  "published": "2023-09-22T18:48:46.000Z",
  "updated": "2023-09-22T18:48:46.000Z",
  "title": "Bootstrap Percolation, Connectivity, and Graph Distance",
  "authors": [
    "Hudson LaFayette",
    "Rayan Ibrahim",
    "Kevin McCall"
  ],
  "comment": "18 pages, 11 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Bootstrap Percolation is a process defined on a graph which begins with an initial set of infected vertices. In each subsequent round, an uninfected vertex becomes infected if it is adjacent to at least $r$ previously infected vertices. If an initially infected set of vertices, $A_0$, begins a process in which every vertex of the graph eventually becomes infected, then we say that $A_0$ percolates. In this paper we investigate bootstrap percolation as it relates to graph distance and connectivity. We find a sufficient condition for the existence of cardinality 2 percolating sets in diameter 2 graphs when $r = 2$. We also investigate connections between connectivity and bootstrap percolation and lower and upper bounds on the number of rounds to percolation in terms of invariants related to graph distance.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-22T18:48:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C12",
      "05C35",
      "05C40"
    ],
    "keywords": [
      "bootstrap percolation",
      "graph distance",
      "connectivity",
      "infected vertices",
      "subsequent round"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}