Results 1  10
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18
Planarizing Graphs  A Survey and Annotated Bibliography
, 1999
"... Given a finite, undirected, simple graph G, we are concerned with operations on G that transform it into a planar graph. We give a survey of results about such operations and related graph parameters. While there are many algorithmic results about planarization through edge deletion, the results abo ..."
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Cited by 32 (0 self)
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Given a finite, undirected, simple graph G, we are concerned with operations on G that transform it into a planar graph. We give a survey of results about such operations and related graph parameters. While there are many algorithmic results about planarization through edge deletion, the results about vertex splitting, thickness, and crossing number are mostly of a structural nature. We also include a brief section on vertex deletion. We do not consider parallel algorithms, nor do we deal with online algorithms.
Enumeration and random realization of triangulated surfaces.arXiv:math.CO
"... We give a complete enumeration of triangulated surfaces with 9 and 10 vertices. Moreover, we discuss how geometric realizations of orientable surfaces with few vertices can be obtained by choosing coordinates randomly. 1 ..."
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Cited by 16 (9 self)
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We give a complete enumeration of triangulated surfaces with 9 and 10 vertices. Moreover, we discuss how geometric realizations of orientable surfaces with few vertices can be obtained by choosing coordinates randomly. 1
Irreducible triangulations of low genus surfaces
 arXiv:math.CO/0606690
"... Abstract. The complete sets of irreducible triangulations are known for the orientable surfaces with genus of 0, 1, or 2 and for the nonorientable surfaces with genus of 1, 2, 3, or 4. By examining these sets we determine some of the properties of these irreducible triangulations. 1. ..."
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Cited by 11 (2 self)
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Abstract. The complete sets of irreducible triangulations are known for the orientable surfaces with genus of 0, 1, or 2 and for the nonorientable surfaces with genus of 1, 2, 3, or 4. By examining these sets we determine some of the properties of these irreducible triangulations. 1.
Isomorphism free lexicographic enumeration of triangulated surfaces and 3manifolds
, 2006
"... We present a fast enumeration algorithm for combinatorial 2 and 3manifolds. In particular, we enumerate all triangulated surfaces with 11 and 12 vertices and all triangulated 3manifolds with 11 vertices. We further determine all equivelar maps on the nonorientable surface of genus 4 as well as a ..."
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Cited by 9 (6 self)
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We present a fast enumeration algorithm for combinatorial 2 and 3manifolds. In particular, we enumerate all triangulated surfaces with 11 and 12 vertices and all triangulated 3manifolds with 11 vertices. We further determine all equivelar maps on the nonorientable surface of genus 4 as well as all equivelar triangulations of the orientable surface of genus 3 and the nonorientable surfaces of genus 5 and 6. 1
Hierarchy of Surface Models and Irreducible Triangulation
 Computational Geometry Theory & Applications
, 2002
"... Given a triangulated closed surface, the problem of constructing a hierarchy of surface models of decreasing level of detail has attracted much attention in computer graphics. A hierarchy provides viewdependent refinement and facilitates the computation of parameterization. For a triangulated cl ..."
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Cited by 9 (0 self)
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Given a triangulated closed surface, the problem of constructing a hierarchy of surface models of decreasing level of detail has attracted much attention in computer graphics. A hierarchy provides viewdependent refinement and facilitates the computation of parameterization. For a triangulated closed surface of n vertices and genus g, we prove that there is a constant c ? 0 such that if n ? c \Delta g, then a greedy strategy can identify \Theta(n) topologypreserving edge contractions that do not interfere with each other. Further, they affect only a constant number of triangles. Repeatedly identifying and contracting such edges produces a topologypreserving hierarchy of O(n + g ) size and O(logn + g) depth. The genus g is very small when compared with n for large models in practice. The identification of edges can be enhanced by selecting edges based on the error of their contractions as measured by some known heuristics.
THE MAXIMUM NUMBER OF CLIQUES IN A GRAPH EMBEDDED IN A Surface
, 2009
"... This paper studies the following question: Given a surface Σ and an integer n, what is the maximum number of cliques in an nvertex graph embeddable in Σ? We characterise the extremal graphs for this question, and prove that the answer is between 8(n − ω) + 2 ω and 8n + 3 2 2ω + o(2 ω), where ω is ..."
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Cited by 6 (2 self)
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This paper studies the following question: Given a surface Σ and an integer n, what is the maximum number of cliques in an nvertex graph embeddable in Σ? We characterise the extremal graphs for this question, and prove that the answer is between 8(n − ω) + 2 ω and 8n + 3 2 2ω + o(2 ω), where ω is the maximum integer such that the complete graph Kω embeds in Σ. For the surfaces S0, S1, S2, N1, N2, N3 and N4 we establish an exact answer.
Surface realization with the intersection edge functional
 arXiv:math.MG/0608538
"... Deciding realizability of a given polyhedral map on a (compact, connected) surface belongs to the hard problems in discrete geometry, from the theoretical, the algorithmic, and the practical point of view. In this paper, we present a heuristic algorithm for the realization of simplicial maps, based ..."
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Cited by 5 (3 self)
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Deciding realizability of a given polyhedral map on a (compact, connected) surface belongs to the hard problems in discrete geometry, from the theoretical, the algorithmic, and the practical point of view. In this paper, we present a heuristic algorithm for the realization of simplicial maps, based on the intersection edge functional. The heuristic was used to find geometric realizations in R 3 for all vertexminimal triangulations of the orientable surfaces of genus g = 3 and g = 4. Moreover, for the first time, examples of simplicial polyhedra in R 3 of genus 5 with 12 vertices were obtained. 1
Irreducible triangulations are small
 J. COMBIN. THEORY SER. B
, 2010
"... A triangulation of a surface is irreducible if there is no edge whose contraction produces another triangulation of the surface. We prove that every irreducible triangulation of a surface with Euler genus g ≥ 1 has at most 13g − 4 vertices. The best previous bound was ..."
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Cited by 5 (0 self)
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A triangulation of a surface is irreducible if there is no edge whose contraction produces another triangulation of the surface. We prove that every irreducible triangulation of a surface with Euler genus g ≥ 1 has at most 13g − 4 vertices. The best previous bound was
Irreducible Triangulations of Surfaces with Boundary
, 2011
"... A triangulation of a surface is irreducible if no edge can be contracted to produce a triangulation of the same surface. In this paper, we investigate irreducible triangulations of surfaces with boundary. We prove that the number of vertices of an irreducible triangulation of a (possibly nonorientab ..."
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Cited by 2 (0 self)
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A triangulation of a surface is irreducible if no edge can be contracted to produce a triangulation of the same surface. In this paper, we investigate irreducible triangulations of surfaces with boundary. We prove that the number of vertices of an irreducible triangulation of a (possibly nonorientable) surface of genus g ≥ 0 with b ≥ 0 boundaries is O(g +b). So far, the result was known only for surfaces without boundary (b = 0). While our technique yields a worse constant in the O(.) notation, the present proof is elementary, and simpler than the previous ones in the case of surfaces without boundary.
Link conditions for simplifying meshes with embedded structures
 IEEE Transactions on Visualization and Computer Graphics
"... Abstract—Interactive visualization applications benefit from simplification techniques that generate good quality coarse meshes from high resolution meshes that represent the domain. These meshes often contain interesting substructures, called embedded structures, and it is desirable to preserve the ..."
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Abstract—Interactive visualization applications benefit from simplification techniques that generate good quality coarse meshes from high resolution meshes that represent the domain. These meshes often contain interesting substructures, called embedded structures, and it is desirable to preserve the topology of the embedded structures during simplification, in addition to preserving the topology of the domain. This paper describes a proof that link conditions, proposed earlier, are sufficient to ensure that edge contractions preserve topology of the embedded structures and the domain. Excluding two specific configurations, the link conditions are also shown to be necessary for topology preservation. Repeated application of edge contraction on an extended complex produces a coarser representation of the domain and the embedded structures. An extension of the quadric error metric is used to schedule edge contractions, resulting in a good quality coarse mesh that closely approximates the input domain and the embedded structures. Index Terms—Embedded structures, extended complex, link conditions, mesh simplification, topology preservation, quadric error metric. F 1