Most of the lectures and lecture notes on Cluster Algebras (at least from Combinatorial point of view) start with mutations of the diagonals of a convex n-gon (mostly the pentagon) as the illustration of creating the new cluster variables using the Ptolemy's theorem, which also manifests other fundamental properties, such as positivity and Laurent phenomenon. (For example this http://arxiv.org/abs/1212.6263.)

Although I imagine one could address this question via the generalization of cluster algebras to other oriented surfaces with boundary, I was wondering if one could start with an arbitrary n-gon (not necessarily convex) and proceed from this point.

For instance, if one just deflates pentagon to an appropriate 5-gon, there are some diagonals without any flips and consequently the number of clusters we get would be different, so the cluster algebra varies and seems to be dependent of the deflation we apply to a fixed n-gon.

Is there any written work on this naive idea of generalization of cluster algebras from convex to arbitrary polygons, approachable for non-experts?

Clear answers and more importantly good references are highly appreciated.