If $G = (V, E)$ is a simple graph with at least one vertex and $G'$ is the graph formed by adding a new vertex $v$ and making it adjacent to every vertex in $V$. How do you show that $G$ has a Hamiltonian path if and only if $G'$ has a Hamiltonian cycle?
If you have a Hamiltonian cycle that means that every vertex is traversed at least once where the graph is linked back to the starting point without backtracking. If the Hamiltonian cycle exists in $G'$ then the exclusion of the $v$ vertex would turn the Hamiltonian cycle into a Hamiltonian path. However, I don't understand how you would prove that using induction or contradiction.