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Irreducible Quadrangulations of the Torus

In this paper, we find the irreducible quadrangulations of the torus. As a consequence, any two quadrangulations of the torus with the same number of vertices that are either both bipartite or both non-bipartite (except for some complete bipartite graphs) can be transformed into one another, up to homeomorphism, using a sequence of diagonal slides and diagonal rotations. We also determine the minor minimal 2-representative graphs on the torus.