  We present the notion of separable game with respect to a forward directed
hypergraph (FDH-graph), which refines and generalizes that of graphical game.
First, we show that there exists a minimal FDH-graph with respect to which a
game is separable, providing a minimal complexity description for the game.
Then, we prove a symmetry property of the minimal FDH-graph of potential games
and we describe how it reflects to a decomposition of the potential function in
terms of local functions. In particular, these last results strengthen the ones
recently proved for graphical potential games. Finally, we study the interplay
between separability and the decomposition of finite games in their harmonic
and potential components, characterizing the separability properties of both
such components.
