General t\^ete-\`a-t\^ete graphs and Seifert manifolds

T^ete-`a-t^ete graphs and relative t^ete-`a-t^ete 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 t^ete-`a-t^ete 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 t^ete-`a-t^ete graph and vice versa.