Let $G$ be a graph of order $n\geq 3$ having property that for every vertex $v$ of $G$, there is a Hamiltonian path with initial vertex $v$. Show that $G$ is $2$-connected but not necessarily Hamiltonian (i.e. needn't contain a Hamiltonian cycle).
The book told me to consider the Petersen graph, but I don't think the Petersen graph is $2$-connected.