Results 1  10
of
24
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 15 (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 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.
Generating irreducible triangulations of surfaces
, 2006
"... Abstract. Starting with the irreducible triangulations of a fixed surface and splitting vertices, all the triangulations of the surface up to a given number of vertices can be generated. The irreducible triangulations have previously been determined for the surfaces S0, S1, N1,and N2. An algorithm i ..."
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Cited by 5 (0 self)
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Abstract. Starting with the irreducible triangulations of a fixed surface and splitting vertices, all the triangulations of the surface up to a given number of vertices can be generated. The irreducible triangulations have previously been determined for the surfaces S0, S1, N1,and N2. An algorithm is presented for generating the irreducible triangulations of a fixed surface using triangulations of other surfaces. This algorithm has been implemented as a computer program which terminates for S1, S2, N1, N2, N3, and N4. Thus the complete sets irreducible triangulations are now also known for S2, N3, and N4, with respective cardinalities 396784, 9708, and 6297982. 1.
kMinimal Triangulations of Surfaces
, 1995
"... A triangulation of a closed surface is kminimal (k 3) if each edge belongs to some essential kcycle and all essential cycles have length at least k. It is proved that the class of kminimal triangulations is finite (up to homeomorphism). As a consequence it follows, without referring to the R ..."
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Cited by 3 (0 self)
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A triangulation of a closed surface is kminimal (k 3) if each edge belongs to some essential kcycle and all essential cycles have length at least k. It is proved that the class of kminimal triangulations is finite (up to homeomorphism). As a consequence it follows, without referring to the RobertsonSeymour's theory, that there are only finitely many minorminimal graph embeddings of given representativity. In the topological part, certain separation properties of homotopic simple closed curves are presented.
Algebraic shifting and fvector theory
, 2007
"... This manuscript focusses on algebraic shifting and its applications to fvector theory of simplicial complexes and more general graded posets. It includes attempts to use algebraic shifting for solving the gconjecture for simplicial spheres, which is considered by many as the main open problem in f ..."
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Cited by 3 (1 self)
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This manuscript focusses on algebraic shifting and its applications to fvector theory of simplicial complexes and more general graded posets. It includes attempts to use algebraic shifting for solving the gconjecture for simplicial spheres, which is considered by many as the main open problem in fvector theory. While this goal has not been achieved, related results of independent interest were obtained, and are presented here. The operator algebraic shifting was introduced by Kalai over 20 years ago, with applications mainly in fvector theory. Since then, connections and applications of this operator to other areas of mathematics, like algebraic topology and combinatorics, have been found by different researchers. See Kalai’s recent survey [34]. We try to find (with partial success) relations between algebraic shifting and the following other areas: • Topological constructions on simplicial complexes. • Embeddability of simplicial complexes: into spheres and other manifolds.
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.
Complexity of triangulations of the projective space
"... It is known that any two triangulations of a compact 3–manifold are related by finite sequences of certain local transformations. We prove here an upper bound for the length of a shortest transformation sequence relating any two triangulations of the 3–dimensional projective space, in terms of the n ..."
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Cited by 1 (1 self)
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It is known that any two triangulations of a compact 3–manifold are related by finite sequences of certain local transformations. We prove here an upper bound for the length of a shortest transformation sequence relating any two triangulations of the 3–dimensional projective space, in terms of the number of tetrahedra.