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BASES OF CLASSES OF TREES CLOSED UNDER SUBDIVISIONS
"... At the Fifth Hungarian Colloquium on Combinatorics held in Keszthely in 1976 L. Lovasz has proposed the following problem [1]: Given two finite graphs G, H,IetG <Hmean that H contains a subdivision of G. Let us call a class JC of finite graphs closed if G e3£, G <H implies HeJC. (For example, ..."
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restrict ourselves to trees. A closed class of trees is defined analogously as a closed class of graphs. Let 3~i> 3~2 be two closed classes of trees, let 381, 38 2 be their bases, respectively. Denote £T = 57~in£T2. A tree T belongs to 3 ~ if and only if it contains simultaneously a subdivision of a
Extension of . . . HigherDegree Parametric Curves  Curve Intersection by the SubdivisionSupercomposition Method
, 2008
"... We present a subdivision algorithm for computing the intersection of spline curves. The complexity depends on geometric quantities that represent the hardness of the computation in a natural way, like the angle of the intersection. The main idea is the application of the supercomposition technique, ..."
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, which considers unions of adjacent parameter intervals that are not siblings in the subdivision tree. This approach addresses the common difficulty of nontermination of the classical subdivision approach when the intersection coincides with a subdivision point, but it avoids the numerical overhead
TREES WHOSE 2DOMINATION SUBDIVISION NUMBER IS 2
, 2012
"... Abstract. A set S of vertices in a graph G = (V, E) is a 2dominating set if every vertex of V \ S is adjacent to at least two vertices of S. The 2domination number of a graph G, denoted by γ2 (G), is the minimum size of a 2dominating set of G. The 2domination subdivision number sdγ 2 (G) is the ..."
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Abstract. A set S of vertices in a graph G = (V, E) is a 2dominating set if every vertex of V \ S is adjacent to at least two vertices of S. The 2domination number of a graph G, denoted by γ2 (G), is the minimum size of a 2dominating set of G. The 2domination subdivision number sdγ 2 (G
Spectra of the subdivisionvertex and subdivisionedge corona
, 2013
"... The subdivision graph S(G) of a graph G is the graph obtained by inserting a new vertex into every edge of G. Let G1 and G2 be two vertex disjoint graphs. The subdivisionvertex corona of G1 and G2, denoted by G1 G2, is the graph obtained from S(G1) and V (G1) copies of G2, all vertexdisjoint, b ..."
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Cited by 3 (1 self)
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The subdivision graph S(G) of a graph G is the graph obtained by inserting a new vertex into every edge of G. Let G1 and G2 be two vertex disjoint graphs. The subdivisionvertex corona of G1 and G2, denoted by G1 G2, is the graph obtained from S(G1) and V (G1) copies of G2, all vertex
Dual complexes of cubical subdivisions
 of R n . Manuscript, IST
, 2010
"... We use a distortion to define the dual complex of a cubical subdivision of R n as an ndimensional subcomplex of the nerve of the set of ncubes. Motivated by the topological analysis of highdimensional digital image data, we consider such subdivisions defined by generalizations of quad and octtr ..."
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Cited by 1 (1 self)
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We use a distortion to define the dual complex of a cubical subdivision of R n as an ndimensional subcomplex of the nerve of the set of ncubes. Motivated by the topological analysis of highdimensional digital image data, we consider such subdivisions defined by generalizations of quad and octtrees
A Subdivision Approach to Maximum Parsimony
 ANNALS OF COMBINATORICS
, 2008
"... Determining an optimal phylogenetic tree using maximum parsimony, also referred to as the Steiner tree problem in phylogenetics, is NP hard. Here we provide a new formulation for this problem which leads to an analytical and linear time solution when the dimensionality (sequence length, or number o ..."
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Cited by 6 (1 self)
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Determining an optimal phylogenetic tree using maximum parsimony, also referred to as the Steiner tree problem in phylogenetics, is NP hard. Here we provide a new formulation for this problem which leads to an analytical and linear time solution when the dimensionality (sequence length, or number
Point location in disconnected planar subdivisions
, 2010
"... Let G be a (possibly disconnected) planar subdivision and let D be a probability measure over R2. The current paper shows how to preprocess (G,D) into an O(n) size data structure that can answer planar point location queries over G. The expected query time of this data structure, for a query point ..."
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Cited by 1 (0 self)
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point drawn according to D, is O(H + 1), where H is a lower bound on the expected query time of any linear decision tree for point location in G. This extends the results of Collette et al. (2008, 2009) from connected planar subdivisions to disconnected planar subdivisions. A version of this structure
Fast HighDimensional Approximation with Sparse Occupancy Trees
, 2010
"... Abstract This paper is concerned with scattered data approximation in high dimensions: Given a data set X ⊂ R d of N data points x i along with values y i ∈ R d , i = 1, . . . , N , and viewing the y i as values y i = f (x i ) of some unknown function f , we wish to return for any query point x ∈ R ..."
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Cited by 94 (9 self)
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and to deal efficiently with large data sets. For this purpose we propose new methods based on what we call sparse occupancy trees and piecewise linear schemes based on simplex subdivisions.
Merging BSP Trees Yields Polyhedral Set Operations
 COMPUTER GRAPHICS
, 1990
"... BSP trees have been shown to provide an effective repretentation of polyhedra through the use of spatial subdivision,;nd are an alternative to the topologically based breps. While?sp tree algorithms are knownfor a number of important opera:ions, such as rendering, no previous work on bsp trees has ..."
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Cited by 100 (2 self)
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BSP trees have been shown to provide an effective repretentation of polyhedra through the use of spatial subdivision,;nd are an alternative to the topologically based breps. While?sp tree algorithms are knownfor a number of important opera:ions, such as rendering, no previous work on bsp trees
Two and ThreeDimensional Point Location in Rectangular Subdivisions
 Journal of Algorithms
, 1995
"... We apply van Emde Boastype stratified trees to point location problems in rectangular subdivisions in 2 and 3 dimensions. In a subdivision with n rectangles having integer coordinates from [0; U \Gamma 1], we locate an integer query point in O((log log U ) d ) query time using O(n) space when d ..."
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Cited by 19 (1 self)
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We apply van Emde Boastype stratified trees to point location problems in rectangular subdivisions in 2 and 3 dimensions. In a subdivision with n rectangles having integer coordinates from [0; U \Gamma 1], we locate an integer query point in O((log log U ) d ) query time using O(n) space when d
Results 11  20
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296