k(G)=is the vertex connectivity
Note: we name the vertex joining to each of v_1,v_2,...,v_k the new vertex.
Proof. By the definition of vertex-connectivity, we need to show that we need to remove at least k vertices from H to make H disconnected.
First of all, by the construction, we know that, if we remove v_1,v_2,...,v_k from H, then the new vertex joining ...
This is a proof regarding a k-connected graph.