Is the complete graph $K_n$ $n-1$-connected?

Question

Is the complete graph $K_n$ $n-1$-connected?

I'm trying to have a better understanding of k-connectedness and it helps to look at an extreme case. My intuition is that it's either that or it's 1-connected.

Answer 1

If we remove $n-2$ vertices from $K_n$ we are left with $2$ connected (all possible edges were present in $K_n$, so this one still is) vertices, which is connected. So by definition $K_n$ is $n-1$ connected; in fact it's the only graph on $n$ vertices that is: if some $(v,w)$, with $v \neq w, v,w \in V$ is not in the edge set $E$ of a graph $G = (V,E)$ we can remove all $n-2$ points $V\setminus\{v,w\}$ from $G$ and then we'd be left with a disconnected $2$-vertex graph. So a graph $(V,E)$ is $(|V|-1)$-connected iff all possible edges are in $E$ already.