Let G be a finite graph. Can G have a subgraph H (H is supposed to be different from G) so that G and H are isomorphic?

Question

Let G be a finite graph. Can G have a subgraph H (H is supposed to be different from G) so that G and H are isomorphic?

Stuck on this question. Don't know how to approach this question. Any answer would be much appreciated!

Answer 1

The subgraph $H$ must have strictly fewer edges than $G$, otherwise $H=G$. But two graphs that are isomorphic must have the same number of edges. Hence $H$ cannot be isomorphic to $G$.

The finiteness restriction is needed here, otherwise $G$ could be an infinite path $0-1-2-3\cdots$ and $H$ could be $1-2-3\cdots$, providing an example.