Let $G(V,E)$ and $H(V',E')$ be two undirected graphs. Suppose $f : V \to V'$ is a function such that $\forall (u,v) \in V\times V, (u,v)\in E \implies (f(u),f(v))\in E' $ and $\forall (u,v) \in V\times V, (u,v)\notin E \implies (f(u),f(v))\notin E' $. In otherwords $f$ preserves edges as well as non-edges of $G$. But it need not be the case that $f$ is injective.
Note that $f$ by definition is a homomorphism from $G$ to $H$. But it is clearly a stricter notion. Is there a name for such morphisms ? And are there results on combinatorics of such morphisms ?