Is there a name for a "hierachical" graph "inferred" from the edges and vertices of another graph?

Solution 1:

By a basic theorem about equivalence relations and projections, a surjective function like "$\mathrm{Category}$" makes a canonical bijection between $C$ and the equivalence classes of the equivalence kernel of the function. In other words, $\mathrm{Category}$ gives rise to a relevant equivalence relation on $V$.

Given such an equivalence relation (which you may interpret as a partition of the vertices into "categories"), then defining the new vertices and edges via your definitions of $V_C$ and $E_C$ is exactly what is done in defining the quotient graph corresponding to that equivalence relation.