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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 8 (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
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 4 (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.
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 2 (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
Decomposition and enumeration of triangulated surfaces
"... We describe some theoretical results on triangulations of surfaces and we develop a theory on roots, decompositions and genussurfaces. We apply this theory to describe an algorithm to list all triangulations of closed surfaces with at most a fixed number of vertices. We specialize the theory for th ..."
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Cited by 1 (0 self)
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We describe some theoretical results on triangulations of surfaces and we develop a theory on roots, decompositions and genussurfaces. We apply this theory to describe an algorithm to list all triangulations of closed surfaces with at most a fixed number of vertices. We specialize the theory for the case where the number of vertices is at most 11 and we get theoretical restrictions on genussurfaces allowing us to get the list of triangulations of closed surfaces with at most 11 vertices.
Surface Realization with the Intersection Segment Functional
, 2009
"... 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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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 segment 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
IsomorphismFree Lexicographic Enumeration of Triangulated Surfaces and 3Manifolds
, 2007
"... 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 polyhedral maps on the nonorientable surface of genus 4 a ..."
Abstract
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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 polyhedral 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.
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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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.