The graph genus problem is NP-complete
Journal of Algorithms
Clique partitions, graph compression and speeding-up algorithms
Journal of Computer and System Sciences
Planar separators and parallel polygon triangulation
Journal of Computer and System Sciences - Special issue on selected papers presented at the 24th annual ACM symposium on the theory of computing (STOC '92)
Efficient computation of implicit representations of sparse graphs
Discrete Applied Mathematics
Geometric compression through topological surgery
ACM Transactions on Graphics (TOG)
A Linear Time Algorithm for Embedding Graphs in an Arbitrary Surface
SIAM Journal on Discrete Mathematics
Linear-Time Succinct Encodings of Planar Graphs via Canonical Orderings
SIAM Journal on Discrete Mathematics
WRAP&Zip decompression of the connectivity of triangle meshes compressed with edgebreaker
Computational Geometry: Theory and Applications - Special issue on multi-resolution modelling and 3D geometry compression
Orderly spanning trees with applications to graph encoding and graph drawing
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Representing dynamic binary trees succinctly
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Low Redundancy in Static Dictionaries with Constant Query Time
SIAM Journal on Computing
A Fast General Methodology for Information-Theoretically Optimal Encodings of Graphs
SIAM Journal on Computing
Edgebreaker: Connectivity Compression for Triangle Meshes
IEEE Transactions on Visualization and Computer Graphics
Compact Encodings of Planar Graphs via Canonical Orderings and Multiple Parentheses
ICALP '98 Proceedings of the 25th International Colloquium on Automata, Languages and Programming
Planarization of Graphs Embedded on Surfaces
WG '95 Proceedings of the 21st International Workshop on Graph-Theoretic Concepts in Computer Science
Succinct representation of balanced parentheses, static trees and planar graphs
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
3D Compression Made Simple: Edgebreaker with Zip&Wrap on a Corner-Table
SMI '01 Proceedings of the International Conference on Shape Modeling & Applications
Compact representations of separable graphs
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
An Information-Theoretic Upper Bound of Planar Graphs Using Triangulation
STACS '03 Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science
Graph compression and the zeros of polynomials
Information Processing Letters
Dissections and trees, with applications to optimal mesh encoding and to random sampling
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Dissections, orientations, and trees with applications to optimal mesh encoding and random sampling
ACM Transactions on Algorithms (TALG)
Quick encoding of plane graphs in log 214 bits per edge
Information Processing Letters
Optimal coding and sampling of triangulations
ICALP'03 Proceedings of the 30th international conference on Automata, languages and programming
A fast and compact web graph representation
SPIRE'07 Proceedings of the 14th international conference on String processing and information retrieval
Succinct representations of separable graphs
CPM'10 Proceedings of the 21st annual conference on Combinatorial pattern matching
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This extended abstract summarizes a new result for the graphcompression problem, addressing how to compress a graphG into a binary string Z with the requirement thatZ can be decoded to recover G. Graphcompression finds important applications in 3D model compression ofComputer Graphics [12, 17-20] and compact routing table of ComputerNetworks [7]. For brevity, let a ¦Ð-graph standfor a graph with property ¦Ð. Theinformation-theoretically optimal number of bits required torepresent an n-node ¦Ð-graph is⌈log2N¦Ð(n)⌉, whereN¦Ð(n) is the number ofdistinct n-node ¦Ð-graphs. Although determiningor approximating the close forms ofN¦Ð(n) for nontrivial classesof ¦Ð is challenging, we provide a linear-timemethodology for graph compression schemes that areinformation-theoretically optimal with respect to continuoussuper-additive functions (abbreviated as optimal for therest of the extended abstract). Specifically, if ¦Ðsatisfies certain properties, then we can compress anyn-node m-edge ¦Ð-graph G into abinary string Z such that G and Z can becomputed from each other in O(m + n) time, and thatthe bit count of Z is at most ¦Â(n) +o(¦Â(n)) for any continuoussuper-additive function ¦Â(n) withlog2N¦Ð(n)¡Ü ¦Â(n) +o(¦Â(n)). Our methodology is applicableto general classes of graphs; this extended abstract focuses ongraphs with sublinear genus. For example, if the inputn-node ¦Ð-graph G is equipped with anembedding on its genus surface, which is a reasonable assumptionfor graphs arising from 3D model compression, then our methodologyis applicable to any ¦Ð satisfying the followingstatements:F1. The genus of any n-node ¦Ð-graph iso(n/log2 n);F2. Any subgraph of a ¦Ð-graph remains a¦Ð-graph;F3. log N ¦Ð(n) =¦¸(n); andF4. There is an integer k = O(1) such that ittakes O(n) time to determine whether anO(log(k) n)-node graph satisfiesproperty ¦Ð.For instance, ¦Ð can be the property of being adirected 3-colorable simple graph with genus no more than ten. Theresult is a novel application of planarization algorithm forbounded-genus graphs [5] and separator decomposition tree of planargraphs [9]. Rooted trees were the only known nontrivial class ofgraphs with linear-time optimal coding schemes. He, Kao, and Lu[11] provided O(n log n)-time compressionschemes for planar and plane graphs that are optimal. Our resultssignificantly enlarge the classes of graphs that admit efficientoptimal compression schemes. More results on various versions ofgraph compression problems or succinct graph representations can befound in [1-4, 6, 8, 10, 14, 15] and the references therein.