Establishing order in planar subdivisions

A planar subdivision is the partition of the plane induced by an embedded planar graph. A representation of such a subdivision isordered if, for each vertexv of the associated graphG, the (say) clockwise sequence of edges in the embedding ofG incident withv appears explicitly.The worst-case complexity of establishing order in a planar subdivision, i.e., converting an unordered representation into an ordered one, is shown to be (n + log (G)), wheren is the size (number of vertices) of the underlying graphG and (G) is (essentially) the number of topologically distinct embeddings ofG in the plane.This work was supported by the National Science and Engineering Research Council of Canada under Grant A3583. A preliminary version of this paper appeared in theProceedings of the Third Annual ACM Symposium on Computational Geometry.