Piping and Instrumentation Diagrams (P&IDs), widely adopted in industrial domains for their high information density, have emerged as a critical research frontier in digitization studies. However, existing methods predominantly focus on the locations of components and connecting lines while neglecting the complex connectivity relationships between components, which can negatively impact downstream applications such as simulation modeling and digital twin construction. Moreover, conventional position-based connectivity recognition approaches exhibit degraded accuracy when handling intricate line structures (e.g., dashed or jumping lines) and dense component layouts. To address these, we propose a graph neural network (GNN)-based framework for components connectivity recognition, which effectively captures component relationships within P&IDs. Additionally, to address the absence of benchmark datasets in this field, we introduce PIDCon, a novel annotated dataset designed for training and evaluating connectivity recognition models. PIDCon also supports end-to-end evaluation of the entire digitization process, including component and line segment detection. Extensive experiments demonstrate that our framework achieves superior performance compared to conventional position-based methods, particularly in scenarios with intricate connectivity patterns. The code and models are available at https://github.com/sad123-yx/PIDCon .