Assembly hinges on reliably forming connections between parts; yet most robotic approaches plan assembly sequences and part poses while treating connectors as an afterthought. Connections represent the foundational physical constraints of assembly execution; while task planning sequences operations, the precise establishment of these constraints ultimately determines assembly success. In this paper, we treat connections as explicit, primary entities in assembly representation, directly encoding connector types, specifications, and locations for every assembly step. Drawing inspiration from how humans learn assembly tasks through step-by-step instruction manuals, we present Manual2Skill++, a vision-language framework that automatically extracts structured connection information from assembly manuals. We encode assembly tasks as hierarchical graphs where nodes represent parts and sub-assemblies, and edges explicitly model connection relationships between components. A large-scale vision-language model parses symbolic diagrams and annotations in manuals to instantiate these graphs, leveraging the rich connection knowledge embedded in human-designed instructions. We curate a dataset containing over 20 assembly tasks with diverse connector types to validate our representation extraction approach, and evaluate the complete task understanding-to-execution pipeline across four complex assembly scenarios in simulation, spanning furniture, toys, and manufacturing components with real-world correspondence. More detailed information can be found at https://nus-lins-lab.github.io/Manual2SkillPP/

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