Does the Collection of Graphs not Embeddable in a Fixed Orbifold have a Well-quasi-ordering?

Question

Using the Robertson-Seymour Theorem, one can show that given a fixed surface $S$ the collection of graphs which can't be embedded in it are defined by a finite set of forbidden minors - just as the non-planar graphs have a finite set ($K_5$ and $K_{3,3}$) of forbidden minors for embeddability.

Is the same true for $2$-dimensional compact orbifolds?

Answer 1

The underlying topological space of a 2-dimensional orbifold is a topological manifold with boundary. The point is that the only allowed stabilizers are cyclic groups of rotations or dihedral groups, which you can calculate model quotients of explicitly.