Definition of Graph Isomorphism

Question

In "Graph Theory" by Reinhard Diestel, the author defines homomorphisms and isomorphisms between graphs as follows:

Let $G=(V,E)$ and $G'=(V', E')$ be two graphs. A map $\varphi:V \to V'$ is a homomorphism if $\varphi(x)\varphi(y)$ is an edge whenever $xy$ is an edge. A homomorphism $\varphi$ is an isomorphism if $\varphi$ is bijective and its inverse is also a homomorphism.

Is there a graph homomorphism $\varphi$ such that $\varphi$ is bijective but $\varphi^{-1}$ is not a homomorphism?

Answer 1

Sure, suppose $G_1$ and $G_2$ each have two vertices, but $G_1$ has no edges and $G_2$ has one edge.

Then a bijection on vertices is a homomorphism from $G_1$ to $G_2$, but not vice versa.