Isomorphism on labeled graphs where the vertices in the same "position" must have equivalent labels? I am looking for information about a specific type of equivalence relation on labeled graphs. Specifically, I want two labeled graphs to be considered equivalent if their underlying unlabeled graphs are isomorphic and vertices in the same "position" have equivalent labels (according to some equivalence relation defined on the vertex labels).

I hope the following example illustrates the type of equivalence relation on labeled graphs that I am looking for.

Example:
The set of possible vertex labels are the letters of the English alphabet, which may be uppercase or lowercase. Now I consider two vertex labels to be equivalent if they are either both an uppercase letter or both a lowercase letter. Now, as an example, consider the following three graphs:

Graph $G_{abC}$:



Graph $G_{deF}$:



Graph $G_{GHi}$:



Let $[a]$ be the equivalence class of lowercase letters and $[A]$ the equivalence class of uppercase letters. If I relabel each vertex with the corresponding equivalence class, then for the first two graphs I obtain the same labeled graph $G_{[a][a][A]}$:



For the third graph, on the other hand, I get a different relabeled graph $G_{[A][A][a]}$:



So although all three graphs are isomorphic, I want to say that only the first two graphs are equivalent. Specifically, $G_{abC}$ and $G_{deF}$ are equivalent because the relabeling with vertex classes yields the same relabeled graph $G_{[a][a][A]}$. The third graph $G_{GHi}$,  on the other hand, is not equivalent to the others because its relabeled graph $G_{[A][A][a]}$ is different from $G_{[a][a][A]}$.

So can anyone tell me if there is a standard name for such a kind of equivalence relation on labeled graphs? Moreover, does anyone know of any references where such an equivalence relation is discussed? In particular, I am interested in algorithms to determine if two labeled graphs are equivalent in the manner I described.
You want to think of your graph vertices as being "weighted" or "colored" (according to your equivalence of labels).  Then when you compute isomorphisms, you want to respect the coloring.  The program nauty has support for this.  The default way this is implemented is that a specific colored vertex can only be sent to a vertex that is the same color, so this should accomplish exactly what you want.
Suppose $G_1$ and $G_2$ are labeled graphs. Let the vertex set of a graph $G=(V,E)$ be $V(G)$ where each $v \in V(G)$ is assumed to have a unique label.

We say that two graphs $G_1$ and $G_2$ are equivalent if there exists an $f(G_1)=(f_v(V_1),f_e(E_1))=G_2$ such that (1) (preservation of adjacency, graph isomorphism) $(v_1,v_2) \in E(G_1) \iff (f_v(v_1),f_v(v_2))\in E(G_2)$ and (2, equivalence) $\forall v \in V(G_1)\; ,\;  f_v(v) \sim v$.

(On the rewrite $\sim$ means equivalent to.)

Also, I have abusively used $f_v(V_1) = \{f_v(v) | v \in V(G_1)\}$

Sorry, if I'm being unclear.