Representation of surface homeomorphisms by tête-à-tête graphs

We use tête-à-tête graphs as defined by N. A'campo and extended versions to codify all periodic mapping classes of an orientable surface with non-empty boundary, improving work of N. A'Campo and C. Graf. We also introduce the notion of mixed tête-à-tête graphs to model some pseudo-periodic homeomorphisms. In particular we are able to codify the monodromy of any irreducible plane curve singularity. The work ends with an appendix that studies all the possible combinatorial structures that make a given filtered metric ribbon graph with some regularity conditions into a mixed tête-à-tête graph