Results 1  10
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15
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 13 (8 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 9 (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 7 (5 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 6 (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.
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
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 1 (0 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.
On Diagrams and Embeddings
, 1997
"... We present methods to automatically compute nice drawings and embeddings of combinatorial objects. As in previous approaches [33, 41] this can be achieved by using a physical model based on spring forces [32] or on electrical forces [10, 21]. This article generalizes these earlier results, since we ..."
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We present methods to automatically compute nice drawings and embeddings of combinatorial objects. As in previous approaches [33, 41] this can be achieved by using a physical model based on spring forces [32] or on electrical forces [10, 21]. This article generalizes these earlier results, since we use a physical model where the edges in a combinatorial complex are represented as springs and a certain number of the vertices has fixed positions or carries electric charges. Also nonlinear spring forces are considered and  in contrast to the previous 2dimensional models  we treat algorithmic solutions for arbitrary dimension d. The minimization of the energy of this system yields in most cases automatically nice drawings and embeddings of combinatorial objects. To have a measure of niceness we define ratios of extreme volumes, e.g., the ratio of the shortest to the longest edge. For some combinatorial objects it is necessary to find a compatible object which has additional structu...
A Note on the Complexity of Real Algebraic Hypersurfaces
"... Given an algebraic hypersurface O in R d, how many simplices are necessary for a simplicial complex isotopic to O? We address this problem and the variant where all vertices of the complex must lie on O. We give asymptotically tight worstcase bounds for algebraic plane curves of degree n. Our resul ..."
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Given an algebraic hypersurface O in R d, how many simplices are necessary for a simplicial complex isotopic to O? We address this problem and the variant where all vertices of the complex must lie on O. We give asymptotically tight worstcase bounds for algebraic plane curves of degree n. Our results gradually improve known bounds in higher dimensions; however, the question for tight bounds remains unsolved for d ≥ 3.
Restricted Edge Contractions in Triangulations of the Sphere with Boundary
"... Given a surface triangulation T of and a subset X of its vertex set V (T), we define a restricted edge contraction as a contraction of an edge connecting X and V (T)−X. Boundary vertices in V (T) − X are only allowed to be contracted to the boundary vertices in X adjacent through boundary edges. In ..."
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Given a surface triangulation T of and a subset X of its vertex set V (T), we define a restricted edge contraction as a contraction of an edge connecting X and V (T)−X. Boundary vertices in V (T) − X are only allowed to be contracted to the boundary vertices in X adjacent through boundary edges. In this paper, we prove that if a triangulation T of the sphere with boundary satisfies some connectivity conditions, then all the vertices in V (T) − X can be merged into X by restricted edge contractions. We also prove that the similar properties hold for a triangulation of the sphere with features. 1