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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 465 (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
Convex Analysis
, 1970
"... In this book we aim to present, in a unified framework, a broad spectrum of mathematical theory that has grown in connection with the study of problems of optimization, equilibrium, control, and stability of linear and nonlinear systems. The title Variational Analysis reflects this breadth. For a lo ..."
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Cited by 5350 (67 self)
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progressed also to the study of socalled stationary points, critical points, and other indications of singularity that a point might have relative to its neighbors, especially in association with existence theorems for differential equations.
THE SHAPE OF RANDOM TANGLEGRAMS
"... Abstract. A tanglegram consists of two binary rooted trees with the same number of leaves and a perfect matching between the leaves of the trees. We show that the two halves of a random tanglegram essentially look like two independently chosen random plane binary trees. This fact is used to derive a ..."
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a number of results on the shape of random tanglegrams, including theorems on the number of cherries and generally occurrences of subtrees, the root branches, the number of automorphisms, and the height. For each of these, we obtain limiting probabilities or distributions. Finally, we investigate
ON THE ENUMERATION OF TANGLEGRAMS AND TANGLED CHAINS
"... Abstract. Tanglegrams are a special class of graphs appearing in applications concerning cospeciation and coevolution in biology and computer science. They are formed by identifying the leaves of two rooted binary trees. We give an explicit formula to count the number of distinct binary rooted tangl ..."
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Abstract. Tanglegrams are a special class of graphs appearing in applications concerning cospeciation and coevolution in biology and computer science. They are formed by identifying the leaves of two rooted binary trees. We give an explicit formula to count the number of distinct binary rooted
The Kuratowski closurecomplement theorem
 New Zealand Journal of Mathematics
"... The Kuratowski ClosureComplement Theorem 1.1. [29] If (X,T) is a topological space and A ⊆ X then at most 14 sets can be obtained from A by taking closures and complements. Furthermore there is a space in which this bound is attained. ..."
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Cited by 6 (0 self)
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The Kuratowski ClosureComplement Theorem 1.1. [29] If (X,T) is a topological space and A ⊆ X then at most 14 sets can be obtained from A by taking closures and complements. Furthermore there is a space in which this bound is attained.
The Kuratowski ClosureComplement Theorem
"... topology, was first posed and proven by the Polish mathematician Kazimierz Kuratowski in 1922. Since then, Kuratowski’s Theorem and its related results, in particular, the structure of the Kuratowski monoid of a topological space, have been the subject of a plethora of papers. The formal statement o ..."
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topology, was first posed and proven by the Polish mathematician Kazimierz Kuratowski in 1922. Since then, Kuratowski’s Theorem and its related results, in particular, the structure of the Kuratowski monoid of a topological space, have been the subject of a plethora of papers. The formal statement
Kuratowski Theorem from below
 Acta SML, 3rd LM Conf
, 2000
"... This note proves the Kuratowski theorem from below, i.e. assuming that Kuratowskilike theorem for FreePlanar graphs is right. Graph is defined as a pair of sets (V, E), where V is the set of vertices and E – the set of edges. For graph G V(G) is its vertex set and E(G) is its edge set. We denote b ..."
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Cited by 1 (1 self)
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This note proves the Kuratowski theorem from below, i.e. assuming that Kuratowskilike theorem for FreePlanar graphs is right. Graph is defined as a pair of sets (V, E), where V is the set of vertices and E – the set of edges. For graph G V(G) is its vertex set and E(G) is its edge set. We denote
Untangling Tanglegrams: Comparing Trees by their Drawings ∗
, 2009
"... A tanglegram is a pair of trees on the same set of leaves with matching leaves in the two trees joined by an edge. Tanglegrams are widely used in biology – to compare evolutionary histories of host and parasite species and to analyze genes of species in the same geographical area. We consider optimi ..."
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Cited by 3 (0 self)
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A tanglegram is a pair of trees on the same set of leaves with matching leaves in the two trees joined by an edge. Tanglegrams are widely used in biology – to compare evolutionary histories of host and parasite species and to analyze genes of species in the same geographical area. We consider
Generalized binary tanglegrams: Algorithms and applications
 In Proc. of BICOP 2009, volume 5462 of LNCS
, 2009
"... Abstract. Several applications require the joint display of two phylogenetic trees whose leaves are matched by intertree edges. This issue arises, for example, when comparing gene trees and species trees or when studying the cospeciation of hosts and parasites. The tanglegram layout problem seeks ..."
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Cited by 3 (0 self)
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we (i) define a generalization of the tanglegram layout problem, called the Generalized Tanglegram Layout (GTL) problem, which allows for arbitrary interconnections between the leaves of the two trees, (ii) provide efficient algorithms for the case when the layout of one tree is fixed, (iii) discuss
KuratowskiType Theorems for Average Genus
 J. Combinatorial Theory B
, 1992
"... Graphs of small average genus are characterized. In particular, a Kuratowskitype theorem is obtained: except for finitely many graphs, a cutedgefree graph has average genus less than or equal to 1 if and only if it is a necklace. We provide a complete list of those exceptions. A Kuratowskitype th ..."
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Cited by 6 (3 self)
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Graphs of small average genus are characterized. In particular, a Kuratowskitype theorem is obtained: except for finitely many graphs, a cutedgefree graph has average genus less than or equal to 1 if and only if it is a necklace. We provide a complete list of those exceptions. A Kuratowski
Results 1  10
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