Summary. 3D meshes are widely used in graphic and simulation applications for approximating 3D objects. When representing complex shapes in a raw data format, meshes consume a large amount of space. Applications calling for compact storage and fast transmission of 3D meshes have motivated the multitude of algorithms developed to efficiently compress these datasets. In this paper we survey recent developments in compression of 3D surface meshes. We survey the main ideas and intuition behind techniques for single-rate and progressive mesh coding. Where possible, we discuss the theoretical results obtained for asymptotic behavior or optimality of the approach. We also list some open questions and directions for future research. 1

Abstract. The family of well-orderly maps is a family of planar maps with the property that every connected planar graph has at least one plane embedding which is a well-orderly map. We show that the number of well-orderly maps with n nodes is at most 2 αn+O(log n) , where α ≈ 4.91. A direct consequence of this is a new upper bound on the number p(n) of unlabeled planar graphs with n nodes, log 2 p(n) � 4.91n. The result is then used to show that asymptotically almost all (labeled or unlabeled), (connected or not) planar graphs with n nodes have between 1.85n and 2.44n edges. Finally we obtain as an outcome of our combinatorial analysis an explicit linear time encoding algorithm for unlabeled planar graphs using, in the worst-case, a rate of 4.91 bits per node and of 2.82 bits per edge. 1

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Abstract. We consider the problem of designing succinct geometric data structures while maintaining efficient navigation operations. A data structure is said succinct if the asymptotic amount of space it uses matches the entropy of the class of structures represented. For the case of planar triangulations with a boundary we propose a succinct representation of the combinatorial information that improves to 2.175 bits per triangle the asymptotic amount of space required and that supports the navigation between adjacent triangles in constant time (as well as other standard operations). For triangulations with m faces of a surface with genus g, our representation requires asymptotically an extra amount of 36(g − 1) lg m bits (which is negligible as long as g ≪ m / lg m). 1

This paper addresses the problem of representing the connectivity information of geometric objects using as little memory as possible. As opposed to raw compression issues, the focus is here on designing data structures that preserve the possibility of answering incidence queries in constant time. We propose in particular the first optimal representations for 3-connected planar graphs and triangulations, which are the most standard classes of graphs underlying meshes with spherical topology. Optimal means that these representations asymptotically match the respective entropy of the two classes, namely 2 bits per edge for 3-connected planar graphs, and 1.62 bits per triangle or equivalently 3.24 bits per vertex for triangulations. These representations support adjacency queries between vertices and faces in constant time.

Abstract. The typical surfaces models handled by contemporary Computer Graphics applications have millions of triangles and numerous connected component, handles and boundaries. Edgebreaker and Spirale Reversi are examples of efficient schemes to compress and decompress their connectivity. A surprisingly simple linear–time implementation has been proposed for triangulated surfaces homeomorphic to a sphere and was subsequently extended to surfaces with handles. Here, we further extend its scope to surfaces with multiple components, handles, and multiple boundaries. The result is a simple and efficient compression/decompression solution for the broad class of orientable manifold surfaces.

ABSTRACT. A 3-manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture posits that every irreducible 3-manifold with infinite fundamental group has a finite cover which is Haken. In this paper, we study random 3-manifolds and their finite covers in an attempt to shed light on this difficult question. In particular, we consider random Heegaard splittings by gluing two handlebodies by the result of a random walk in the mapping class group of a surface. For this model of random 3-manifold, we are able to compute the probabilities that the resulting manifolds have finite covers of particular kinds. Our results contrast with the analogous probabilities for groups coming from random balanced presentations, giving quantitative theorems to the effect that 3-manifold groups have many more finite quotients than random groups. The next natural question is whether these covers have positive betti number. For abelian covers of a fixed type over 3-manifolds of Heegaard genus 2, we show that the probability of positive betti number is 0. In fact, many of these questions boil down to questions about the mapping class group. We are lead to consider the action of mapping class group of a surface Σ on

Abstract. This article introduces new algorithms for the uniform random generation of labelled planar graphs. Its principles rely on Boltzmann samplers, as recently developed by Duchon, Flajolet, Louchard, and Schaeffer. It combines the Boltzmann framework, a suitable use of rejection, a new combinatorial bijection found by Fusy, Poulalhon and Schaeffer, as well as a precise analytic description of the generating functions counting planar graphs, which was recently obtained by Giménez and Noy. This gives rise to an extremely efficient algorithm for the random generation of planar graphs. There is a preprocessing step of some fixed small cost; and the expected time complexity of generation is quadratic for exact-size uniform sampling and linear for approximate-size sampling. This greatly improves on the best previously known time complexity for exact-size uniform sampling of planar graphs with n vertices, which was a little over O(n 7). This is the extended and revised journal version of a conference paper with the title “Quadratic exact-size and linear approximate-size random generation of planar graphs”, which appeared in the Proceedings of the International Conference on Analysis of Algorithms (AofA’05), 6-10 June 2005, Barcelona. 1.

Abstract — Previous mesh compression techniques provide decent properties such as high compression ratio, progressive decoding, and out-of-core processing. However, only a few of them supports the random accessibility in decoding, which enables the details of any specific part to be available without decoding other parts. This paper proposes an effective framework for the random accessibility of mesh compression. The key component of the framework is a wire-net mesh constructed from a chartification of the given mesh. Charts are compressed separately for random access to mesh parts and a wire-net mesh provides an indexing and stitching structure for the compressed charts. Experimental results show that random accessibility can be achieved with competent compression ratio, which is only a little worse than single-rate and comparable to progressive encoding. To demonstrate the merits of the framework, we apply it to process huge meshes in an out-of-core manner, such as outof-core rendering and out-of-core editing. Index Terms — Random accessible compression, Mesh chartification, Out-of-core mesh processing

The Baxter number Bn can be written as Bn = � n 0 Θk,n−k−1 with Θk,ℓ = 2 (k + 1) 2 (k + 2)