k-vertex-connected graph

Remarks

A graph is $1$-connected if and only if it is connected and has at least $2$ vertices. A $k$-connected graph is also $j$-connected for each $j \le k$. Example: the cycle graph $C_n$ with $n \ge 3$ is $2$-connected: it has $n > 2$ vertices, and removing one vertex leaves a path, which is connected. It is not $3$-connected, since removing two non-adjacent vertices disconnects it when $n \ge 4$, and $C_3$ has only $3$ vertices. Non-example: a tree with at least $3$ vertices is not $2$-connected, because removing a vertex that is not a leaf disconnects it. The complete graph $K_n$ has $\kappa(K_n) = n - 1$; clause 1 is what prevents it from being $n$-connected. Every $k$-connected graph has minimum degree at least $k$.