Proof By Contradiction - Hamiltonian Paths and Cycles
Contradiction isn't the most natural way to do this, since there's a straightforward constructive proof. 

Suppose $G$ has a Hamiltonian path.  How can you extend that to a Hamiltonian cycle of&nbsp;$G'$?  Conversely, suppose that $G'$&nbsp;has a Hamiltonian cycle. Cover up $v$ with your finger&nbsp;&ndash; what do you see?

It's possible to rewrite the reasoning hinted at above as a proof by contradiction, but it's not very natural. ("Suppose $G$&nbsp;has a Hamiltonian path but $G'$&nbsp;has no Hamiltonian cycle. Oh, look, here's a Hamiltonian cycle in&nbsp;$G'$&nbsp;&ndash; contradiction.")