Discrete Minimal 2-manifolds
We can call a 2-manifold (a finite simple graph for which all unit spheres are circular graphs with 4 or more elements) minimal, if no edge refinement can make it smaller. It is an interesting question to explore what the minima are. In the sphere case, there is only one minimum, the octahedon graph. The other surface types show more minima. How many are there? What is the range of the possible vertex sets?
An other thing bothers me currently: is there an elegant way to see the classification of 2-manifolds without invoking the continuum. Topology books use the geometric realization. There should be a canonical way to deform, possibly with additional Pochner moves to realize a small version as a disk on which different parts of the boundary are identified.