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NearOptimal Connectivity Encoding of 2Manifold Polygon Meshes
, 2002
"... ... this paper we introduce a connectivity encoding method which extends these ideas to 2manifold meshes consisting of faces with arbitrary degree. The encoding algorithm exploits duality by applying valence enumeration to both the primal and dual mesh in a symmetric fashion. It generates two sequen ..."
Abstract

Cited by 54 (6 self)
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... this paper we introduce a connectivity encoding method which extends these ideas to 2manifold meshes consisting of faces with arbitrary degree. The encoding algorithm exploits duality by applying valence enumeration to both the primal and dual mesh in a symmetric fashion. It generates two sequences of symbols, vertex valences and face degrees, and encodes them separately using two contextbased arithmetic coders. This allows us to exploit vertex and/or face regularity if present. When the mesh exhibits perfect face regularity (e.g., a pure triangle or quad mesh) and/or perfect vertex regularity (valence six or four respectively) the corresponding bit rate vanishes to zero asymptotically. For triangle meshes, our technique is equivalent to earlier valence driven approaches. We report compression results for a corpus of standard meshes. In all cases we are able to show coding gains over earlier coders, sometimes as large as 50%. Remarkably, we even slightly gain over coders specialized to triangle or quad meshes. A theoretical analysis reveals that our approach is nearoptimal as we achieve the Tutte entropy bound for arbitrary planar graphs of 2 bits per edge in the worst case.
Orderly Spanning Trees with Applications to Graph Encoding and Graph Drawing
 In 12 th Symposium on Discrete Algorithms (SODA
, 2001
"... The canonical ordering for triconnected planar graphs is a powerful method for designing graph algorithms. This paper introduces the orderly pair of connected planar graphs, which extends the concept of canonical ordering to planar graphs not required to be triconnected. Let G be a connected planar ..."
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Cited by 36 (6 self)
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The canonical ordering for triconnected planar graphs is a powerful method for designing graph algorithms. This paper introduces the orderly pair of connected planar graphs, which extends the concept of canonical ordering to planar graphs not required to be triconnected. Let G be a connected planar graph. We give a lineartime algorithm that obtains an orderly pair (H
The number of polyhedral (3connected planar ) graphs
 Mathematics of Computation 37 (1981), 523–532. CMP 95:17
"... Abstract. Data is presented on the number of 3connected planar graphs, isomorphic to the graphs of convex polyhedra, with up to 26 edges. Results have been checked with the the number of rooted cnets of R.C. Mullin and P.J. Schellenberg and Liu Yanpei. 1. ..."
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Cited by 3 (1 self)
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Abstract. Data is presented on the number of 3connected planar graphs, isomorphic to the graphs of convex polyhedra, with up to 26 edges. Results have been checked with the the number of rooted cnets of R.C. Mullin and P.J. Schellenberg and Liu Yanpei. 1.