Once you are in the setting where vertices have types (which usually would be called "colours" in the graph literature), there is little point in having edges with missing endpoints - just use introduce a new colour for vertices that are "not really there".

Now the situation that vertices in a graph are identified according to some rules seems to be pretty common occurrance. E.g. a simple system of the kind you are interested in could be described in the following way using a more typical graph theory language:

Let G, H be a graph where the vertices are coloured red, green and white. Write $G \preceq H$ if there are two vertices $v, u$ in $H$ such that $v$ is coloured red, $u$ is coloured green and that $G$ is obtained from $H$ by identifying $v$ and $u$ and colouring the resulting vertex white.

Question: Given some particular coloured graph $H$, what are the graphs $G \preceq H$ coloured completely white?

I'm not aware of any good overview on such approaches, but hopefully such a reformulation can help you finding the kind of results you are looking for.