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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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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.
Bridges between Geometry and Graph Theory
 in Geometry at Work, C.A. Gorini, ed., MAA Notes 53
"... Graph theory owes many powerful ideas and constructions to geometry. Several wellknown families of graphs arise as intersection graphs of certain geometric objects. Skeleta of polyhedra are natural sources of graphs. Operations on polyhedra and maps give rise to various interesting graphs. Another ..."
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Cited by 9 (4 self)
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Graph theory owes many powerful ideas and constructions to geometry. Several wellknown families of graphs arise as intersection graphs of certain geometric objects. Skeleta of polyhedra are natural sources of graphs. Operations on polyhedra and maps give rise to various interesting graphs. Another source of graphs are geometric configurations where the relation of incidence determines the adjacency in the graph. Interesting graphs possess some inner structure which allows them to be described by labeling smaller graphs. The notion of covering graphs is explored.
Reconfigurations of polygonal structures
, 2005
"... This thesis contains new results on the subject of polygonal structure reconfiguration. Specifically, the types of structures considered here are polygons, polygonal chains, triangulations, and polyhedral surfaces. A sequence of vertices (points), successively joined by straight edges, is a polygona ..."
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Cited by 8 (1 self)
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This thesis contains new results on the subject of polygonal structure reconfiguration. Specifically, the types of structures considered here are polygons, polygonal chains, triangulations, and polyhedral surfaces. A sequence of vertices (points), successively joined by straight edges, is a polygonal chain. If the sequence is cyclic, then the object is a polygon. A planar triangulation is a set of vertices with a maximal number of noncrossing straight edges joining them. A polyhedral surface is a threedimensional structure consisting of flat polygonal faces that are joined by common edges. For each of these structures there exist several methods of reconfiguration. Any such method must provide a welldefined way of transforming one instance of a structure to any other. Several types of reconfigurations are reviewed in the introduction, which is followed by new results. We begin with efficient algorithms for comparing monotone chains. Next, we prove that flat chains with unitlength edges and angles within a wide range always admit reconfigurations, under the dihedral model of motion. In this model, angles and edge lengths are preserved. For the universal
The Number of Embeddings of Minimally Rigid Graphs
 GEOMETRY © 2003 SPRINGERVERLAG NEW YORK INC.
, 2003
"... Rigid frameworks in some Euclidean space are embedded graphs having a unique local realization (up to Euclidean motions) for the given edge lengths, although globally they may have several. We study the number of distinct planar embeddings of minimally rigid graphs with n vertices. We show that, mo ..."
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Cited by 7 (2 self)
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Rigid frameworks in some Euclidean space are embedded graphs having a unique local realization (up to Euclidean motions) for the given edge lengths, although globally they may have several. We study the number of distinct planar embeddings of minimally rigid graphs with n vertices. We show that, modulo planar rigid motions, this number is at most � � 2n−4 n ≈ 4. We also exhibit several families which realize lower bounds n−2 of the order of 2n,2.21n and 2.28n. For the upper bound we use techniques from complex algebraic geometry, based on the (projective) Cayley–Menger variety CM2,n (C) ⊂ P n ( 2)−1(C) over the complex numbers C. In this context, point configurations are represented by coordinates given by squared distances between all pairs of points. Sectioning the variety with 2n − 4 hyperplanes yields at most deg(CM2,n) zerodimensional components, and one finds this degree to be D2,n � � = 1 2n−4. The lower bounds are related to inductive constructions of minimally rigid graphs 2 n−2 via Henneberg sequences. The same approach works in higher dimensions. In particular, we show that it leads to an upper bound of 2D3,n = (2n−3 /(n − 2)) � � 2n−6 for the number of spatial embeddings n−3 with generic edge lengths of the 1skeleton of a simplicial polyhedron, up to rigid motions. Our technique can also be adapted to the nonEuclidean case.
OnLine Convex Planarity Testing
, 1995
"... An important class of planar straightline drawings of graphs are the convex drawings, in which all faces are drawn as convex polygons. A graph is said to be convex planar if it admits a convex drawing. We consider the problem of testing convex planarity in a semidynamic environment, where a graph i ..."
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Cited by 6 (3 self)
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An important class of planar straightline drawings of graphs are the convex drawings, in which all faces are drawn as convex polygons. A graph is said to be convex planar if it admits a convex drawing. We consider the problem of testing convex planarity in a semidynamic environment, where a graph is subject to online insertions of vertices and edges. We present online algorithms for convex planarity testing with the following performance, where t denotes the number of vertices of the graph: convex planarity testing and insertion of vertices take 0(1) worstcase tinhe, insertion of edges takes 0(log n) amortized tinhe, and the space requirement of the data structure is O(n). Furthermore, we give a new combinatorial characterization of convex planar graphs.
Polytope Skeletons And Paths
 Handbook of Discrete and Computational Geometry (Second Edition ), chapter 20
"... INTRODUCTION The kdimensional skeleton of a dpolytope P is the set of all faces of the polytope of dimension at most k. The 1skeleton of P is called the graph of P and denoted by G(P ). G(P ) can be regarded as an abstract graph whose vertices are the vertices of P , with two vertices adjacent i ..."
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INTRODUCTION The kdimensional skeleton of a dpolytope P is the set of all faces of the polytope of dimension at most k. The 1skeleton of P is called the graph of P and denoted by G(P ). G(P ) can be regarded as an abstract graph whose vertices are the vertices of P , with two vertices adjacent if they form the endpoints of an edge of P . In this chapter, we will describe results and problems concerning graphs and skeletons of polytopes. In Section 17.1 we briefly describe the situation for 3polytopes. In Section 17.2 we consider general properties of polytopal graphs subgraphs and induced subgraphs, connectivity and separation, expansion, and other properties. In Section 17.3 we discuss problems related to diameters of polytopal graphs in connection with the simplex algorithm and t
M.: Many Triangulated 3Spheres
, 2002
"... We construct 2 Ω(n5/4) combinatorial types of triangulated 3spheres on n vertices. Since by a result of Goodman and Pollack (1986) there are no more than 2 O(n log n) combinatorial types of simplicial 4polytopes, this proves that asymptotically, there are far more combinatorial types of triangulat ..."
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We construct 2 Ω(n5/4) combinatorial types of triangulated 3spheres on n vertices. Since by a result of Goodman and Pollack (1986) there are no more than 2 O(n log n) combinatorial types of simplicial 4polytopes, this proves that asymptotically, there are far more combinatorial types of triangulated 3spheres than of simplicial 4polytopes on n vertices. This complements results of Kalai (1988), who had proved a similar statement about dspheres and (d + 1)polytopes for fixed d ≥ 4.
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
Computational Polygonal Entanglement Theory
 In VIII Encuentros de Geometría Computacional
, 1999
"... In this paper we are concerned with motions for untangling polygonal linkages (chains, polygons and trees) in 2 and 3 dimensions. We say that an open, simple polygonal chain can be straightened if it can be continuously reconfigured to a sequence of collinear segments in such a way that the rigidity ..."
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Cited by 5 (3 self)
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In this paper we are concerned with motions for untangling polygonal linkages (chains, polygons and trees) in 2 and 3 dimensions. We say that an open, simple polygonal chain can be straightened if it can be continuously reconfigured to a sequence of collinear segments in such a way that the rigidity and length of each link and the simplicity of the entire chain are maintained throughout the motion. For a closed chain (simple polygon) untangling means convexification: reconfiguration to a convex polygon. For a tree untangling means "flattening". Linkages that cannot be untangled are called locked. Whether a simple open chain in 2D can be straightened remains a tantalizing open problem. For some special classes of chains it is known that they can be straightened. On the other hand a tree can lock. In 3D both open and closed chains can lock without being knotted. An open chain can be straightened if it has a simple orthogonal projection onto some plane. Furthermore, a planar closed simple...
On Touching Triangle Graphs
"... Abstract. In this paper, we consider the problem of representing graphs by triangles whose sides touch. We present linear time algorithms for creating touching triangles representations for outerplanar graphs, square grid graphs, and hexagonal grid graphs. The class of graphs with touching triangles ..."
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Abstract. In this paper, we consider the problem of representing graphs by triangles whose sides touch. We present linear time algorithms for creating touching triangles representations for outerplanar graphs, square grid graphs, and hexagonal grid graphs. The class of graphs with touching triangles representations is not closed under minors, making characterization difficult. We do show that pairs of vertices can only have a small common neighborhood, and we present a complete characterization of the subclass of biconnected graphs that can be represented as triangulations of some polygon.