The paper explores the evolution of cooperation in structured populations using a public goods game (PGG) with a generalized interaction mode. Unlike traditional models where players interact with all their neighbors, this study introduces a scenario where individuals select a subset of their neighbors to form groups for PGG interactions. The authors use a pair-approximation approach and weak-selection limits to analyze how the number of total neighbors and the size of the restricted group influence the critical enhancement factor required for cooperation to dominate over defection. The results show that the critical enhancement factor is lower in this restricted interaction mode compared to the traditional setup, indicating better conditions for the evolution of cooperation. The theoretical findings are supported by numerical simulations, which confirm the effectiveness of the proposed interaction mode in promoting cooperation. The study also discusses the implications of different update rules and population structures on the evolutionary dynamics of cooperation.The paper explores the evolution of cooperation in structured populations using a public goods game (PGG) with a generalized interaction mode. Unlike traditional models where players interact with all their neighbors, this study introduces a scenario where individuals select a subset of their neighbors to form groups for PGG interactions. The authors use a pair-approximation approach and weak-selection limits to analyze how the number of total neighbors and the size of the restricted group influence the critical enhancement factor required for cooperation to dominate over defection. The results show that the critical enhancement factor is lower in this restricted interaction mode compared to the traditional setup, indicating better conditions for the evolution of cooperation. The theoretical findings are supported by numerical simulations, which confirm the effectiveness of the proposed interaction mode in promoting cooperation. The study also discusses the implications of different update rules and population structures on the evolutionary dynamics of cooperation.