$G$ is 2-vertex-connected graph if and only if every 2 vertices from $G$ lie on a simple cycle.

Question

$G$ is $2$-vertex-connected graph if and only if every $2$ vertices from $G$ lie on a simple cycle.

$\implies$

If $G$ is $2$-vertex connected graph it means for every $ v \in V(G), \deg(v) \ge 2 $ because if exist any $ v_1 $ and $ \deg v_1 = 1 $ then we can erase neighboring vertex and graph will not connected so won't be 2-vertex connected. So if $ v \in V(G), \deg(v) \le 2 $ then it exist cycle. $ \deg_v \ge 2 $ so if we start in $v$ we have to back to this vertex.

$\impliedby$ I don't know.

Answer 1

I think your forwards implication is a little vague.

Take two vertices $u$ and $v$. We want to find a simple cycle containing them. As you've noted, each vertex must have at least two neighbours. So let $u_1$ and $u_2$ be two neighbours of $u$, and consider the graph we get from deleting $u$. We know by assumption that it is still connected, so we can find paths from $u_1$ to $v$ and $u_2$ to $v$. Joining these together will give us a simple cycle containing $u$ and $v$, so we're done — provided the two paths don't intersect. Can every such path have an intersection? Explain why not.

For the reverse implication, take a vertex $x$ which we want to delete, and two vertices $a$ and $b$ which we want to connect in the resulting graph. Using the fact that we can conjure up a simple cycle containing $a$ and $b$, show that we can get a path in $G \setminus \{ x \}$ containing $a$ and $b$.

Answer 2

There is an easier way to justify the forward implication.

We know that with the help of Menger's Theorem, that if a graph is 2-vertex connected, then there exists two disjoint paths for any two vertices u and v of the vertex set of the Graph. If we choose any two vertices in the graph and pick a path P(u,v) and Q(v,u), then we have a cycle given by [P(u,v)Q(v,u)] that contains u and v.