{
  "id": "1904.06527",
  "version": "v1",
  "published": "2019-04-13T11:14:21.000Z",
  "updated": "2019-04-13T11:14:21.000Z",
  "title": "On the $g$-extra connectivity of graphs",
  "authors": [
    "Zhao Wang",
    "Yaping Mao",
    "Sun-Yuan Hsieh"
  ],
  "comment": "20 pages; 2 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Connectivity and diagnosability are two important parameters for the fault tolerant of an interconnection network $G$. In 1996, F\\`{a}brega and Fiol proposed the $g$-extra connectivity of $G$. A subset of vertices $S$ is said to be a \\emph{cutset} if $G-S$ is not connected. A cutset $S$ is called an \\emph{$R_g$-cutset}, where $g$ is a non-negative integer, if every component of $G-S$ has at least $g+1$ vertices. If $G$ has at least one $R_g$-cutset, the \\emph{$g$-extra connectivity} of $G$, denoted by $\\kappa_g(G)$, is then defined as the minimum cardinality over all $R_g$-cutsets of $G$. In this paper, we first obtain the exact values of $g$-extra connectivity of some special graphs. Next, we show that $1\\leq \\kappa_g(G)\\leq n-2g-2$ for $0\\leq g\\leq \\left\\lfloor \\frac{n-3}{2}\\right\\rfloor$, and graphs with $\\kappa_g(G)=1,2,3$ and trees with $\\kappa_g(T_n)=n-2g-2$ are characterized, respectively. In the end, we get the three extremal results for the $g$-extra connectivity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-13T11:14:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "extra connectivity",
      "interconnection network",
      "extremal results",
      "important parameters",
      "fault tolerant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}