In this paper we consider cooperative transferable utility games with limited communication structure, called graph games. Agents are able to cooperate only if they can communicate directly or indirectly with each other. For the class of acyclic graph games the average tree solution has recently been proposed. It was proven that the average tree solution is a core element if the game exhibits super-additivity. We show that the condition of super-additivity can be relaxed to a weaker condition, which admits for a natural interpretation. Moreover, we introduce the concept of subcore, which is a subset of the core, always contains the average tree solution, and therefore is a non-empty refinement of the core.