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" (and if you want to distinguish different kinds of edges, give the virtual vertices different colours).

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.