The intuition is that each edge of the ICG defines a path in the class graph. We use the compatability concept in two stages. At the instantiation level we check that the concrete class graph is compatible with the interface class graph. A refined strategy is compatible with the non-refined strategy, according to the definitions. It is useful to define equivalence between strategy graphs.
Equivalence holds if refinement holds in both directions, possibly with different name maps.
We need a simple example, where G1 is an expansion of G2 but G1 is not a refinement of G2. G1 is not a refinement of G2. But G1 is an expansion of G2. There is a connection between no-surprise strategies and graph refinement. The paper talks only about three relations.