Abstract: In this paper we introduce fast and efficient inverse subdivision algorithms, with linear time and space complexity, to detect and reconstruct uniform Loop, Catmull-Clark, and Doo-Sabin subdivision structure in irregular triangular, quadrilateral, and polygonal meshes. We consider two main applications for these algorithms. The first one is to enable interactive modeling systems that support uniform subdivision surfaces to use popular interchange file formats which do not preserve the subdivision structure, such as VRML, without loss of information. The second application is to improve the compression efficiency of existing lossless connectivity compression schemes, by optimally compressing meshes with Loop subdivision connectivity. Our Loop inverse subdivision algorithm is based on global connectivity properties of the covering mesh, a concept motivated by the covering surface from Algebraic Topology. Although the same approach can be used for other subdivision schemes, such as Catmull-Clark, we present a Catmull-Clark inverse subdivision algorithm based on a much simpler graph-coloring algorithm and a Doo-Sabin inverse subdivision algorithm based on properties of the dual mesh. Straightforward extensions of these approaches to other popular uniform subdivision schemes are also discussed.