A question about simple connected graph such that all degrees are at least $2$

Question

Let $G$ be A simple connected graph such that all degrees are at least $2$. Is it true that every vertex is on a circle ?

Answer 1

It's not true. A counter example would be two disjoint cycles plus one connecting edge. The connecting edge is not part of any cycle.

Edit: To get a vertex that is not on a cylce one can modify the example above by putting an extra vertex in the middle of the connecting edge.