Yin-Yang Puzzles are NP-complete

We prove NP-completeness of Yin-Yang / Shiromaru-Kuromaru pencil-and-paper puzzles. Viewed as a graph partitioning problem, we prove NP-completeness of partitioning a rectangular grid graph into two induced trees (normal Yin-Yang), or into two induced connected subgraphs (Yin-Yang without $2\times 2$ rule), subject to some vertices being pre-assigned to a specific tree/subgraph.