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Heterogeneous integration of single-crystal materials offers great opportunities for advanced device platforms and functional systems1. Although substantial efforts have been made to co-integrate active device layers by heteroepitaxy, the mismatch in lattice polarity and lattice constants has been limiting the quality of the grown materials2. Layer transfer methods as an alternative approach, on the other hand, suffer from the limited availability of transferrable materials and transfer-process-related obstacles3. Here, we introduce graphene nanopatterns as an advanced heterointegration platform that allows the creation of a broad spectrum of freestanding single-crystalline membranes with substantially reduced defects, ranging from non-polar materials to polar materials and from low-bandgap to high-bandgap semiconductors. Additionally, we unveil unique mechanisms to substantially reduce crystallographic defects such as misfit dislocations, threading dislocations and antiphase boundaries in lattice- and polarity-mismatched heteroepitaxial systems, owing to the flexibility and chemical inertness of graphene nanopatterns. More importantly, we develop a comprehensive mechanics theory to precisely guide cracks through the graphene layer, and demonstrate the successful exfoliation of any epitaxial overlayers grown on the graphene nanopatterns. Thus, this approach has the potential to revolutionize the heterogeneous integration of dissimilar materials by widening the choice of materials and offering flexibility in designing heterointegrated systems.