I'm interested in triangulations with few vertices of a given orientable compact surface $S$.
By triangulation, I don't mean a "simplicial triangulation" but a "decomposition of $S$ by triangles", these being topological triangles glued by identifying edges. The only condition required is that an edge in $S$ is adjacent to two different triangles.
For instance, gluing two copies of a given euclidean triangle along the corresponding edges gives what I consider to be a triangulation of the sphere with 3 vertices (and 3 edges and 2 faces).
Question 1: the study of such "triangulations" must be classical. What are good references?
More specifically, I'm interested in the following question:
Question 2: for $S$ of low genus (say 0, 1 or 2) what are the different combinatorial types of such triangulations with few (say 2,3 or 4) vertices? And what is the method to describe them?
Thanks in advance for any help.