General tête-à-tête graphs and Seifert manifolds

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Tête-à-tête graphs and relative tête-à-tête graphs were introduced by N. A'Campo in 2010 to model monodromies of isolated plane curves. By recent workof Fdez de Bobadilla, Pe Pereira and the author, they provide a way of modeling the periodic mapping classes that leave some boundary component invariant. In this work we introduce the notion of general tête-à-tête graph and prove that they model all periodic mapping classes. We also describe algorithms that take a Seifert manifold and a horizontal surface and return a tête-à-tête graph and vice versa.