{
  "id": "2208.06324",
  "version": "v1",
  "published": "2022-08-12T15:26:08.000Z",
  "updated": "2022-08-12T15:26:08.000Z",
  "title": "On the Connectivity and Diameter of Geodetic Graphs",
  "authors": [
    "Asaf Etgar",
    "Nati Linial"
  ],
  "comment": "10 pages, 5 fugures",
  "categories": [
    "math.CO"
  ],
  "abstract": "A graph $G$ is geodetic if between any two vertices there exists a unique shortest path. In 1962 Ore raised the challenge to characterize geodetic graphs, but despite many attempts, such characterization still seems well beyond reach. We may assume, of course, that $G$ is $2$-connected, and here we consider only graphs with no vertices of degree $1$ or $2$. We prove that all such graphs are, in fact $3$-connected. We also construct an infinite family of such graphs of the largest known diameter, namely $5$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-12T15:26:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "connectivity",
      "unique shortest path",
      "characterize geodetic graphs",
      "characterization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}