This study investigates the evolution of cooperation in structured public goods games (PGG) with a generalized interaction mode. In traditional PGG models, individuals interact with all their neighbors, but in this model, individuals select only a subset of their neighbors to form a group for interaction. The research explores how the number of total neighbors and the size of the restricted group influence the critical enhancement factor, which determines whether cooperation dominates over defection. Using a pair-approximation approach in the weak-selection limit, the study theoretically analyzes the stability of the system and shows that the critical enhancement factor is lower in this generalized interaction mode compared to the traditional setup. This suggests that the generalized interaction mode provides a better condition for the evolution of cooperation. Numerical simulations support these findings, demonstrating that cooperation can emerge when the enhancement factor exceeds a critical threshold. The study also shows that the critical enhancement factor increases with the number of neighbors and the group size, making cooperation harder to evolve. However, the critical value in the generalized interaction mode is always smaller than in the traditional model, indicating that this mode is more favorable for cooperation. The results highlight the importance of interaction mode in shaping the evolution of cooperation in structured populations.This study investigates the evolution of cooperation in structured public goods games (PGG) with a generalized interaction mode. In traditional PGG models, individuals interact with all their neighbors, but in this model, individuals select only a subset of their neighbors to form a group for interaction. The research explores how the number of total neighbors and the size of the restricted group influence the critical enhancement factor, which determines whether cooperation dominates over defection. Using a pair-approximation approach in the weak-selection limit, the study theoretically analyzes the stability of the system and shows that the critical enhancement factor is lower in this generalized interaction mode compared to the traditional setup. This suggests that the generalized interaction mode provides a better condition for the evolution of cooperation. Numerical simulations support these findings, demonstrating that cooperation can emerge when the enhancement factor exceeds a critical threshold. The study also shows that the critical enhancement factor increases with the number of neighbors and the group size, making cooperation harder to evolve. However, the critical value in the generalized interaction mode is always smaller than in the traditional model, indicating that this mode is more favorable for cooperation. The results highlight the importance of interaction mode in shaping the evolution of cooperation in structured populations.