Let $G = (V,E)$ be an undirected finite simple graph with no loops.
Definition Let's call a subset $U\subset V$ autonomous if every vertex of $V\ \setminus U$ is either adiacent to every vertex of $U$ or not adiacent to any of them.
This notion is used in J. Körner, G. Simonyi, Zs. Tuza, Perfect couples of graphs, Combinatorica, 12 (2) (1992), 179-192 to derive an equation that graph entropy must satisfy. This equation relates the entropy of $G$ with the entropy of the smaller graph obtained by collapsing $U$ to a single vertex, connected with each vertex that was connected to every vertex of $U$.
I was surprised to learn in chat that a similar construction is used in automata theory to simplify certain finite state machines.
Are there other constructions that are usually employed to link a certain function on a graph with the same function calculated on a smaller but related graph?
[My motivation is to conjecture their equivalent for graph entropy and try to prove them.]