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

T\^ete-\`a-t\^ete twists, monodromies and representation of elements of Mapping Class Group