Assume we have a connected Graph G with more than 2 vertices and which is 2-vertex-connected. Let $u,v \in V$ and $u \neq v$. I want to prove that there exists a cycle in G which contains $u$ and $v$.
Since it is 2-vertex-connected it is possible to find 2 vertex-independent paths connecting these vertices( Menger's theorem). This 2 vertex-independent paths forms together a cycle. Is this proof correct and sufficient?
What about the reverse implication? A clique with 6 vertex is a counterexample. Since there is a circle for each vertex pair that contains these vertex, but the graph is not 2-vertex-connected or ?