  We introduce a concept of minimality for Fano polygons. We show that, up to
mutation, there are only finitely many Fano polygons with given singularity
content, and give an algorithm to determine the mutation-equivalence classes of
such polygons. This is a key step in a program to classify orbifold del Pezzo
surfaces using mirror symmetry. As an application, we classify all Fano
polygons such that the corresponding toric surface is qG-deformation-equivalent
to either (i) a smooth surface; or (ii) a surface with only singularities of
type 1/3(1,1).
