Organized research on topological graph theory centered around re-embeddings ofgraphs

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By computer experiments, we fixed errors in the classification of panel structures on the torus already published, and found an example of re-embedding of a triangulation on the orientable closed surface of genus 2 that does not decompose into elementary parts. As applications of re-embedding theory, we gave a sufficient condition for two independent edges in a graph on a closed surface to extend to a perfect matching, and specified the structures on concrete surface forbidding such extension. We also showed the existence of an upper bound for the distinguishing numbers of triangulations on a closed surface and gave the least upper bounds for the projective plane and the torus.