Let $G=(V,E)$ be a simple $2$-connected graph and $C$ is a cycle in $G$ satisfies:

For any vertex $v$ of $C$,there exists at least one vertex $u\in V(G)\backslash V(C)$ adjacent with $v$.

Is it ture that there must exists a cycle in $G$ which is longer than $C$?

Let $G=(V,E)$ be a simple $2$-connected graph and $C$ is a cycle in $G$ satisfies:

For any vertex $v$ of $C$,there exists at least one vertex $u\in V(G)\backslash V(C)$ adjacent with $v$.

Is it true that there must exists a cycle in $G$ which is longer than $C$?