Contributed a theoretical enhancement and documentation update to the OI-wiki/OI-wiki repository, focusing on planar graph min-cut transformations. Developed a formal theorem specifying the conditions under which the minimum cut of a planar graph can be mapped to the shortest path in its dual, using rigorous algorithm analysis and graph theory principles. Supplemented the theorem with a counterexample to address common misconceptions, thereby improving the accuracy and depth of the documentation. Enhanced the clarity of technical explanations by refining wording throughout the relevant sections. Work was conducted using LaTeX and Markdown, emphasizing precise documentation and knowledge sharing for the community.