For every vertex of a Graph G, k(G-v) = k(G) or k(G-v) = k(g) -1?

Question

can you help me with my homework ?

"Prove or desprove : Let $G $ be a nontrivial Graph. For every vertex of a Graph $G$, $k(G-v) = k(G)$ or $k(G-v) = k(g) -1$ "

I think the answer is : "Once we are removing only one vertex, we can only have two cases, 1) this vertex $v$ is a member of the lowest cut vertex, 2) this vertex $v$ is not a member of the cut vertex.

In the case 1) the removal of $v$ will not alter the cutvertex $(k)$. 2) the removal of $v$ will decrease the cutvertex in $1$.

So, if we only have this two cases, we can prove that is true. "

But I dont know if it is right. Can you help me? Thanks :)

Answer 1

This theorem is false. Take the graph below, call it $G$.

It can be seen that $k(G) = 2$, but $k(G - v) = k(K_4) = 3$.