  Inspired by the seminal result that a graph and an associated rotation system
uniquely determine the topology of a closed manifold, we propose a
combinatorial method for reconstruction of surfaces from points. Our method
constructs a spanning tree and a rotation system. Since the tree is trivially a
planar graph, its rotation system determines a genus zero surface with a single
face which we proceed to incrementally refine by inserting edges to split faces
and thus merging them. In order to raise the genus, special handles are added
by inserting edges between different faces and thus merging them. We apply our
method to a wide range of input point clouds in order to investigate its
effectiveness, and we compare our method to several other surface
reconstruction methods. We find that our method offers better control over
outlier classification, i.e. which points to include in the reconstructed
surface, and also more control over the topology of the reconstructed surface.
