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71
Rotation distance, triangulations, and hyperbolic geometry
 J. Amer. Math. Soc
, 1988
"... A rotation in a binary tree is a local restructuring of the tree that changes it into another tree. One can execute a rotation by collapsing an internal edge of the tree to a point, thereby obtaining a node with three children, and then reexpanding the node of order three in the alternative way int ..."
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Cited by 110 (4 self)
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A rotation in a binary tree is a local restructuring of the tree that changes it into another tree. One can execute a rotation by collapsing an internal edge of the tree to a point, thereby obtaining a node with three children, and then reexpanding the node of order three in the alternative way into two nodes of
Art gallery and illumination problems
 In Handbook on Computational Geometry, Elsevier Science Publishers, J.R. Sack and
, 2000
"... How many guards are necessary, and how many are sufficient to patrol the paintings and works of art in an art gallery with n walls? This wonderfully naïve question of combinatorial geometry has, since its formulation, stimulated an increasing number of of papers and surveys. In 1987, J. O’Rourke pub ..."
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Cited by 86 (3 self)
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How many guards are necessary, and how many are sufficient to patrol the paintings and works of art in an art gallery with n walls? This wonderfully naïve question of combinatorial geometry has, since its formulation, stimulated an increasing number of of papers and surveys. In 1987, J. O’Rourke published his book Art Gallery Theorems and Algorithms which has further fueled this area of research. The present book is being written almost 10 years since the publication of O’Rourke’s book, and the need for an uptodate manuscript on Art Gallery or Illumination Problems is evident. Some important open problems stated in O’Rourke’s book, such as... have been solved. New directions of research have since been investigated, including: watchman routes, floodlight illumination problems, guards with limited visibility or mobility, illumination of families of convex sets on the plane, guarding of rectilinear polygons, and others. In this book, we study these results and try to give a complete
The Cycle Space of an Infinite Graph
 COMB., PROBAB. COMPUT
, 2004
"... Finite graph homology may seem trivial, but for infinite graphs things become interesting. We present a new approach that builds the cycle space of a graph not on its finite cycles but on its topological circles, the homeomorphic images of the unit circle in the space formed by the graph togethe ..."
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Cited by 26 (9 self)
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Finite graph homology may seem trivial, but for infinite graphs things become interesting. We present a new approach that builds the cycle space of a graph not on its finite cycles but on its topological circles, the homeomorphic images of the unit circle in the space formed by the graph together with its ends. Our approach
4connected projectiveplanar graphs are Hamiltonian
 J. Combin. Theory Ser. B
, 1994
"... We prove the result stated in the title (conjectured by Grünbaum), and a conjecture of Plummer that every graph which can be obtained from a 4–connected planar graph by deleting two vertices is Hamiltonian. The proofs are constructive and give rise to polynomial–time algorithms. 2 1. ..."
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Cited by 24 (9 self)
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We prove the result stated in the title (conjectured by Grünbaum), and a conjecture of Plummer that every graph which can be obtained from a 4–connected planar graph by deleting two vertices is Hamiltonian. The proofs are constructive and give rise to polynomial–time algorithms. 2 1.
Finding Hamiltonian Cycles in Delaunay Triangulations Is NPComplete
 IN PROC. 4TH CANAD. CONF. COMPUT. GEOM
, 1994
"... It is shown that it is an NPcomplete problem to determine whether a Delaunay triangulation or an inscribable polyhedron has a Hamiltonian cycle. It is also shown that there exist nondegenerate Delaunay triangulations and simplicial, inscribable polyhedra without 2factors. ..."
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Cited by 19 (1 self)
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It is shown that it is an NPcomplete problem to determine whether a Delaunay triangulation or an inscribable polyhedron has a Hamiltonian cycle. It is also shown that there exist nondegenerate Delaunay triangulations and simplicial, inscribable polyhedra without 2factors.
Greedy Drawings of Triangulations
, 2007
"... Greedy Routing is a class of routing algorithms in which the packets are forwarded in a manner that reduces the distance to the destination at every step. In an attempt to provide theoretical guarantees for a class of greedy routing algorithms, Papadimitriou and Ratajczak [PR05] came up with the fol ..."
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Cited by 19 (1 self)
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Greedy Routing is a class of routing algorithms in which the packets are forwarded in a manner that reduces the distance to the destination at every step. In an attempt to provide theoretical guarantees for a class of greedy routing algorithms, Papadimitriou and Ratajczak [PR05] came up with the following conjecture: Any 3connected planar graph can be drawn in the plane such that for every pair of vertices s and t a distance decreasing path can be found. A path s = v1,v2,...,vk = t in a drawing is said to be distance decreasing if �vi − t � < �vi−1 − t�, 2 ≤ i ≤ k where �... � denotes the Euclidean distance. We settle this conjecture in the affirmative for the case of triangulations. A partitioning of the edges of a triangulation G into 3 trees, called the realizer of G, was first developed by Walter Schnyder who also gave a drawing algorithm based on this. We generalize Schnyder’s algorithm to obtain a whole class of drawings of any given triangulation G. We show, using the KnasterKuratowskiMazurkiewicz Theorem, that some drawing of G belonging to this class is greedy. 1 1
GraphTheoretical Conditions for Inscribability and Delaunay Realizability
 Proceedings of the 6th Canadian Conference on Computational Geometry
, 1995
"... We present new graphtheoretical conditions for polyhedra of inscribable type and Delaunay triangulations. We establish several sufficient conditions of the following general form: if a polyhedron has a sufficiently rich collection of Hamiltonian subgraphs, then it is of inscribable type. These resu ..."
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Cited by 16 (3 self)
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We present new graphtheoretical conditions for polyhedra of inscribable type and Delaunay triangulations. We establish several sufficient conditions of the following general form: if a polyhedron has a sufficiently rich collection of Hamiltonian subgraphs, then it is of inscribable type. These results have several consequences: ffl All 4connected polyhedra are of inscribable type. ffl All simplicial polyhedra in which all vertex degrees are between 4 and 6, inclusive, are of inscribable type. ffl All triangulations without chords or nonfacial triangles are realizable as combinatorially equivalent Delaunay triangulations. We also strengthen some earlier results about matchings in polyhedra of inscribable type. Specifically, we show that any nonbipartite polyhedron of inscribable type has a perfect matching containing any specified edge, and that any bipartite polyhedron of inscribable type has a perfect matching containing any two specified disjoint edges. We give examples showing t...
Smarandache MultiSpace Theory
, 2011
"... Our WORLD is a multiple one both shown by the natural world and human beings. For example, the observation enables one knowing that there are infinite planets in the universe. Each of them revolves on its own axis and has its own seasons. In the human society, these rich or poor, big or small countr ..."
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Cited by 14 (5 self)
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Our WORLD is a multiple one both shown by the natural world and human beings. For example, the observation enables one knowing that there are infinite planets in the universe. Each of them revolves on its own axis and has its own seasons. In the human society, these rich or poor, big or small countries appear and each of them has its own system. All of these show that our WORLD is not in homogenous but in multiple. Besides, all things that one can acknowledge is determined by his eyes, or ears, or nose, or tongue, or body or passions, i.e., these six organs, which means the WORLD consists of have and not have parts for human beings. For thousands years, human being has never stopped his steps for exploring its behaviors of all kinds. We are used to the idea that our space has three dimensions: length, breadth and height with time providing the fourth dimension of spacetime by Einstein. In the string or superstring theories, we encounter 10 dimensions. However, we do not even know what the right degree of freedom is, as Witten said. Today, we have known two heartening notions for sciences. One is the Smarandache multispace came into being by purely logic.
Geometric Graph Theory
, 1999
"... A geometric path is a graph drawn in the plane such that its vertices are points in general position and its edges... This paper surveys some Turántype and Ramseytype extremal problems for geometric graphs, and discusses their generalizations and applications. ..."
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Cited by 12 (0 self)
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A geometric path is a graph drawn in the plane such that its vertices are points in general position and its edges... This paper surveys some Turántype and Ramseytype extremal problems for geometric graphs, and discusses their generalizations and applications.
Exact solution of a class of frequency assignment problems in cellular networks and other regular grids
 in: 8th Italian Conf. Theor. Comp. Sci. (ICTCS’03), LNCS
, 2003
"... For any non negative real values h and k, an L(h, k)labeling of a graph G = (V, E) is a function L: V → IR such that L(u) − L(v)  ≥ h if (u, v) ∈ E and L(u) − L(v)  ≥ k if there exists w ∈ V such that (u, w) ∈ E and (w, v) ∈ E. The span of an L(h, k)labeling is the difference between th ..."
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Cited by 12 (5 self)
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For any non negative real values h and k, an L(h, k)labeling of a graph G = (V, E) is a function L: V → IR such that L(u) − L(v)  ≥ h if (u, v) ∈ E and L(u) − L(v)  ≥ k if there exists w ∈ V such that (u, w) ∈ E and (w, v) ∈ E. The span of an L(h, k)labeling is the difference between the largest and the smallest value of L, so it is not restrictive to assume 0 as the smallest value of L. We denote by λh,k(G) the smallest real λ such that graph G has an L(h, k)labeling of span λ. The aim of the L(h, k)problem is to satisfy the distance constraints using the minimum span. In this paper, we study L(h, k)labeling problem on regular grids of degree 3, 4, 6, and 8 solving several open problems left in the literature. Keywords: L(h,k)labeling, triangular grids, hexagonal grids, squared grids, octagonal grids. 1