Notation of graph with vertices & edges which belong to classes?

Question

I am not well-versed in maths, and know very little about maths notation.

In my research I'm defining a graph, whose vertices and edges can belong to one of a number of defined classes, and I'm wondering whether there is notation to express this.

For example, the nodes V in G can belong to one of the classes {a,b,c}, and the edges can belong to one of the classes {x,y,z}. How can I define the graph in notation, e.g. $G = (V_{\{a,b,c\}}, E_{\{x,y,z\}})$?

Answer 1

We usually say that the vertices are of 3 types, and describe them. The same with the edges. There is no reason to use a $V_{\{a,b,c\}}$ sort of notation, you just define $V$ as consisting of three types of objects.

Answer 2

I don't know of any such notation. You should invent one that's easy for your readers to parse and define it clearly with examples. One possibility is to label each vertex with the class it's in, so something like

Let $\mathcal{V}$ be a set of (disjoint) sets we will use for vertices of a graph $G = (V, E)$. Define $L: V \to \mathcal{V}$ by setting $L(v) = S$ when $v \in S \in \mathcal{V}$.

Whether this turns out to be convenient or bulky depends on how you use the information in your arguments.