Symmetries of Unlabelled Planar Triangulations

We derive a decomposition scheme of unlabelled triangulations rooted at a single cell, where the decomposition depends on whether the automorphism group of the triangulation contains reflections, rotations, or both. Furthermore, the decomposition scheme is constructive in the sense that for each of the three cases, there is a $k\in\mathbb{N}$ such that the scheme defines a one-to-$k$ correspondence between the respective triangulations and their decompositions.