I just learned about line graphs $L(G)$ such that all vertices $v$ in $L(G)$ represent edges in $G$.
Is there a way to inverse this process and for any graph $G$ find $G'$ such that $G$ represents the line graph $L(G)$ of $G'$?
The potential usage of this would be that you could find G' and test it for Eulerianism and then therefore have a algorithm for Hamiltonian cycles. Because of that I assume this must not be possible. If so I would be interested in why it isn't possible?