Add to ORKGAbstract. In this paper we study cooperative transferable utility games with communi-cation structure represented by an undirected graph, i.e., a group of players can cooperate only if they are connected on the graph. This type of games is called graph games and the best-known solution for them is the Myerson value, characterized by component ef-ficiency and fairness. Recently on cycle-free graph games, the average tree solution has been proposed and it is characterized by component efficiency and component fairness. We propose!-parameterized fairness that incorporates the preceding fairness axioms on cycle-free graph games and show the existence and the uniqueness of a solution satisfying component efficiency and our fairness for any nonne...