Let S be a compact orientable surface. Let A and B be two subsurfaces of S that have the same signature. How to check if there is a homeomorphism of S that sends A to B? 

Here a subsurface of $S$ is a component of S \ simple closed curves.

I saw this question https://mathoverflow.net/questions/283567/mapping-class-groups-of-subsurfaces-of-a-hyperbolic-surface and https://arxiv.org/pdf/math/9906122.pdf but I didn't find the part related to my question yet.