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  "body": "\n## Definition\n\nLet $k$ be a positive integer and let $G = (V, E)$ be a finite [[graph|0JD5C82QMB]].\nThe graph $G$ is $k$-vertex-connected, or $k$-connected, if:\n\n1. $V$ has more than $k$ elements.\n2. For each set $X \\subseteq V$ with fewer than $k$ elements, the [[induced subgraph|0V05FC4GKQ]] $G - X$ on $V \\setminus X$ is [[connected|0FFAYNMAJZ]].\n\nThe vertex connectivity $\\kappa(G)$ is the largest integer $k$ such that $G$ is $k$-connected, and $\\kappa(G) = 0$ if there is no such $k$.\n\n## Remarks\n\nA graph is $1$-connected if and only if it is connected and has at least $2$ vertices.\nA $k$-connected graph is also $j$-connected for each $j \\le k$.\nExample: the [[cycle graph|0QQ0F12962]] $C_n$ with $n \\ge 3$ is $2$-connected: it has $n \u003e 2$ vertices, and removing one vertex leaves a path, which is connected.\nIt is not $3$-connected, since removing two non-adjacent vertices disconnects it when $n \\ge 4$, and $C_3$ has only $3$ vertices.\nNon-example: a [[tree|0MJ0JS72DZ]] with at least $3$ vertices is not $2$-connected, because removing a vertex that is not a leaf disconnects it.\nThe [[complete graph|03WKB2H474]] $K_n$ has $\\kappa(K_n) = n - 1$; clause 1 is what prevents it from being $n$-connected.\nEvery $k$-connected graph has minimum degree at least $k$.\n",
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