  We study monodromies of plane curve singularities and pseudo-periodic
homeomorphisms of oriented surfaces with boundary, following an original idea
of the first author: t\^ete-\`a-t\^ete graphs and twists. We completely
characterize mapping classes that can be represented by t\^ete-\`a-t\^ete
twists, and generalize the notion to be able to represent any class of the
mapping class group relative to the boundary which is boundary-free periodic.
This improves previous work on the subject by C. Graf. Furthermore, we
introduce the class of mixed t\^ete-\`a-t\^ete graphs and twists, and prove
that mixed t\^ete-\`a-t\^ete twists contain monodromies of irreducible plane
curve singularities. In a sequel paper, the fourth author and B. Sigurdsson
have extended this to the reducible case.
