{
  "id": "2306.15301",
  "version": "v1",
  "published": "2023-06-27T08:39:47.000Z",
  "updated": "2023-06-27T08:39:47.000Z",
  "title": "Connectivity of 2-distance graphs",
  "authors": [
    "S. H. Jafari",
    "S. R. Musawi"
  ],
  "comment": "7 pages, 6 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "For a simple graph $G$, the $2$-distance graph, $D_2(G)$, is a graph with the vertex set $V(G)$ and two vertices are adjacent if and only if their distance is $2$ in the graph $G$. In this paper, we characterize all graphs with connected 2-distance graph. For graphs with diameter 2, we prove that $D_2(G)$ is connected if and only if $G$ has no spanning complete bipartite subgraphs. For graphs with a diameter greater than 2, we define a maximal Fine set and by contracting $G$ on these subsets, we get a new graph $\\hat G$ such that $D_2(G)$ is connected if and only if $D_2(\\hat G)$ is connected. Especially, $D_2(G)$ is disconnected if and only if $\\hat G$ is bipartite.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-06-27T08:39:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C12",
      "05C15",
      "G.2.2"
    ],
    "keywords": [
      "connectivity",
      "spanning complete bipartite subgraphs",
      "maximal fine set",
      "vertex set",
      "simple graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}