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A Separator Theorem for Planar Graphs
, 1977
"... Let G be any nvertex planar graph. We prove that the vertices of G can be partitioned into three sets A, B, C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains more than 2n/3 vertices, and C contains no more than 2& & vertices. We exhibit an algorithm which ..."
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Cited by 461 (1 self)
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Let G be any nvertex planar graph. We prove that the vertices of G can be partitioned into three sets A, B, C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains more than 2n/3 vertices, and C contains no more than 2& & vertices. We exhibit an algorithm which finds such a partition A, B, C in O(n) time.
Applications of Geometric Separator Theorems
, 1998
"... The companion paper "Geometric separator theorems" proved a large number of separator theorems about geometrical objects. We now find geometric separator theorems about geometrical graphs, and applications (mostly algorithmic) of them. Examples: I: There exists a rectangle crossed by the m ..."
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Cited by 3 (0 self)
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The companion paper "Geometric separator theorems" proved a large number of separator theorems about geometrical objects. We now find geometric separator theorems about geometrical graphs, and applications (mostly algorithmic) of them. Examples: I: There exists a rectangle crossed
Geometric Separator Theorems
, 1998
"... We have a 4step method for producing "separator theorems" about geometrical objects en masse. Examples: I: Given N disjoint isooriented squares in the plane, there exists a rectangle with 2N=3 squares inside, 2N=3 squares outside, and (4 + o(1)) p N partly in & out. II: More gen ..."
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We have a 4step method for producing "separator theorems" about geometrical objects en masse. Examples: I: Given N disjoint isooriented squares in the plane, there exists a rectangle with 2N=3 squares inside, 2N=3 squares outside, and (4 + o(1)) p N partly in & out. II: More
An Upward Measure Separation Theorem
 Theoretical Computer Science
, 1991
"... It is shown that almost every language in ESPACE is very hard to approximate with circuits. It follows that P<Fnan> 6= BPP implies that E is a measure 0 subset of ESPACE. 1 Introduction Hartmanis and Yesha [HY84] proved that P is a proper subset of P/Poly " PSPACE if and only if E is ..."
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Cited by 8 (6 self)
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is a proper subset of ESPACE. (See section 2 for notation and terminology used in this introduction.) This refined the downward separation result E ae 6= ESPACE =) P ae 6= PSPACE of Book [Boo74] and also led immediately to the upward separation result P ae 6= BPP =) E ae 6= ESPACE (1
SEPARATION THEOREMS FOR COMPACT HAUSDORFF FOLIATIONS
, 812
"... Abstract. We investigate compact Hausdorff foliations on compact Riemannian manifolds in the context of the GromovHausdorff distance theory. We give some sufficient conditions for such foliations to be separated in the GromovHausdorff topology (GHseparation theorem). 1. ..."
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Abstract. We investigate compact Hausdorff foliations on compact Riemannian manifolds in the context of the GromovHausdorff distance theory. We give some sufficient conditions for such foliations to be separated in the GromovHausdorff topology (GHseparation theorem). 1.
THE JORDANBROUWER SEPARATION THEOREM
"... Abstract. The Classical Jordan Curve Theorem says that every simple closed curve in R2 divides the plane into two pieces, an “inside ” and an “outside ” of the curve. This paper will prove an considerable extension of this Theorem; that, in fact, every compact, connected hypersurface in Rn divides R ..."
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Abstract. The Classical Jordan Curve Theorem says that every simple closed curve in R2 divides the plane into two pieces, an “inside ” and an “outside ” of the curve. This paper will prove an considerable extension of this Theorem; that, in fact, every compact, connected hypersurface in Rn divides
Application of A Planar Separator Theorem
"... Through the paper Application of A Planar Separator Theorem by RICHARD J. LIPTON and ROBERT TAR JAN, we get that any nvertex planar graph can be divided into components of roughly equal size by removing only O(x/) vertices. This separator theorem with a divideandconquer strategy can help us reso ..."
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Through the paper Application of A Planar Separator Theorem by RICHARD J. LIPTON and ROBERT TAR JAN, we get that any nvertex planar graph can be divided into components of roughly equal size by removing only O(x/) vertices. This separator theorem with a divideandconquer strategy can help us
Geometric Separator Theorems & Applications
"... We find a large number of "geometric separator theorems" such as: I: Given N disjoint isooriented squares in the plane, there exists a rectangle with 2N=3 squares inside, 2N=3 squares outside, and (4 + o(1)) p N partly in & out. II: There exists a rectangle that is crossed by the ..."
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Cited by 16 (0 self)
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We find a large number of "geometric separator theorems" such as: I: Given N disjoint isooriented squares in the plane, there exists a rectangle with 2N=3 squares inside, 2N=3 squares outside, and (4 + o(1)) p N partly in & out. II: There exists a rectangle that is crossed
Separation Theorems for simplicity 26surfaces ⋆
"... Abstract. The main goal of this paper is to prove a Digital Jordan– Brouwer Theorem and an Index Theorem for simplicity 26surfaces. For this, we follow the approach to Digital Topology introduced in [2], and find a digital space such that the continuous analogue of each simplicity 26surface is a c ..."
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combinatorial 2manifold. Thus, the separation theorems quoted above turn out to be an immediate consequence of the general results obtained in [2] and [3] for arbitrary digital nmanifolds. Key words: Digital surface, simplicity 26surface, digital separation theorems.
Duality and separation theorems in idempotent semimodules
 Linear Algebra and its Applications 379 (2004), 395–422. Also arXiv:math.FA/0212294
"... Abstract. We consider subsemimodules and convex subsets of semimodules over semirings with an idempotent addition. We introduce a nonlinear projection on subsemimodules: the projection of a point is the maximal approximation from below of the point in the subsemimodule. We use this projection to sep ..."
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Cited by 70 (22 self)
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to separate a point from a convex set. We also show that the projection minimizes the analogue of Hilbert’s projective metric. We develop more generally a theory of dual pairs for idempotent semimodules. We obtain as a corollary duality results between the row and column spaces of matrices with entries
Results 1  10
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2,967