I know there are things known as homomorphisms in graph theory, which map adjacent vertices to adjacent vertices. However, are there "surjective homomorphisms" or epimorphisms (I'm not quite sure if this is used correctly), that in essence, map adjacent vertices to several adjacent vertices.
For example, we could define map between a graph $G$ and $H$ where if two vertices $u,v$ are adjacent in $G$, then there are 4 vertices $m_1,m_2,n_1,n_2$ forming a $C_4$ subgraph in $H$. More specifically, $u$ maps to $m_1,m_2$ and $v$ maps to $n_1,n_2$.
What this mapping would create would be a graph with many cycles replacing adjacent vertices.
E.g, if we applied that mapping to a star graph on $4$ vertices (including the center), then we would have two vertices that are adjacent in the center and 3 pairs of vertices that replaced the tips of the star to form $C_4$'s with the center two vertices.
The above may be a bad example and mapping, but is this kind of mapping a well defined idea in graph theory? If so, is there a more formal term and outline of this kind of surjective homomorphism? Or has it never really been done before?