Minimal dessins and their deformations

For dessins d'enfants of a certain family (arbitrary dessins, pure dessins, (2,3)-dessins) we propose to consider sets of minimal dessins d'enfants: dessins from this family of genus g with the minimum possible number of edges. In other words, we consider single-cell dessins d'enfants with the passports - (2g+1 | 2g+1 | 2g+1) - (4g | 2^2g | 4g) - (3^(4g-2) | 2^(6g-3) | 6(2g-1)) The number of minimal dessins d'enfants grows very quickly with g, but additional invariants (symmetry group, chameleon group, edge rotation group) split them into many different orbits. We will try to understand minimal dessins d'enfants of genus g<=4, and also consider some Fried families that pass through Belyi pairs corresponding to these dessins d'enfants.