Dessins d'enfants on reducible surfaces

Grothendieck dessins d'enfants are connected embedded graphs of certain special structure on smooth oriented compact surfaces without the boundary. They are naturally connected with do-called Belyi pairs, i.e., non-constant meromorphic functions with at most 3 critical values defined on algebraic curves. Generalized Grothendieck dessins d'enfants, are not necessary connected embedded graphs on not necessary smooth surfaces. Generalized Grothendieck dessins correspond to Belyi function on reducible algebraic curves. We are going to formally introduce and investigate these notions and various types of relations between them: algebraic, geometric, category theory. We will discuss several applications (or even origins) of generalized dessins d'enfants. The talk is based on joint works with N. Ya. Amburg and G. B. Shabat.