Question about isomorphism between two graphs

Question

If two graphs are isomorphic then does there always exist one vertex $u\in A,v\in B$ such that removal of this vertices from respective graphs still gives isomorphism? I tried to find contradiction by example with no success.Then I think it is possible every time by picking maximum degree vertex from both and remove it. Am I right?

Answer 1

Let $f:V_A\to V_B$ be a bijection between the vertices of $A$ and those of $B$. Because this is an isomorphism, any subset of vertices $A'\in V_A$ induces a subgraph isomorphic to that induced by $B'=f(A')\in V_B$. In the case where $A'$ contains all but one vertex, the desired conclusion is obtained.