Establishing order in planar subdivisions

Abstract

A planar subdivision is the partition of the plane induced by an embedded planar graph. A representation of such a subdivision is ordered if, for each vertex v of the associated graph G, the (say) clockwise sequence of edges in the embedding of G incident with v 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 Θ(n + log λ (G)), where n is the size (number of vertices) of the underlying graph G and λ (G) is (essentially) the number of topologically distinct embeddings of G in the plane. © 1988 Springer-Verlag New York Inc.