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  "body": "\n## Summary\n\nA graph is $k$-vertex-connected, or $k$-connected, if it has more than $k$ vertices and stays [[connected|0KWFFMDF8T]] after the removal of any fewer than $k$ vertices.\nThe largest such $k$ is the vertex connectivity $\\kappa(G)$, a measure of how robust a network is against the failure of nodes.\n[[Menger's theorem|E6GYGZDBS7]] gives another view.\nA graph with more than $k$ vertices is $k$-connected if and only if every two vertices are joined by $k$ internally disjoint paths.\nThe node [[k-vertex-connected graph|JJN72VZCZE]] gives the precise definition.\n\n## Explanations\n\n### L1 human\n\nThink of a computer network in which some machines may break down.\nThe network is $2$-connected if the rest of the network can still talk after any single machine fails.\nA ring of machines is $2$-connected: removing one machine leaves a chain, which is still in one piece.\nA chain is only $1$-connected, since removing a machine in the middle cuts it in two.\n",
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  "front": {
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  "msc": [
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  "name": "k-vertex-connected graph",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/7FZ/7FZB7YYBXA.md",
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  "url": "https://math2.tamnd.com/wiki/k-vertex-connected-graph/",
  "writing": {
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        "what": "an inventory row",
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