# How many graphs can have the same line graph?

Suppose we have a finite simple graph $G$. Call $\mathcal O$ the set of graphs without isolated vertices up to isomorphism whose line graph is isomorphic to $G$. Can $\mathcal O$ contain more than one element?

Whitney's result is given as, "If the line graphs of two connected graphs are isomorphic, then the underlying graphs are isomorphic, except in the case of the triangle graph $K_3$ and the claw $K_{1,3}$, which have isomorphic line graphs but are not themselves isomorphic."