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26
QuasiPlanar Graphs Have a Linear Number of Edges
, 1995
"... A graph is called quasiplanar if it can be drawn in the plane so that no three of its edges are pairwise crossing. It is shown that the maximum number of edges of a quasiplanar graph with n vertices is O(n). 1 Introduction We say that an undirected graph G(V; E) without loops or parallel edges is ..."
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Cited by 36 (17 self)
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A graph is called quasiplanar if it can be drawn in the plane so that no three of its edges are pairwise crossing. It is shown that the maximum number of edges of a quasiplanar graph with n vertices is O(n). 1 Introduction We say that an undirected graph G(V; E) without loops or parallel edges is drawn in the plane if each vertex v 2 V is represented by a distinct point and each edge e = (u; v) 2 E is Work on this paper by Pankaj K. Agarwal, Boris Aronov and Micha Sharir has been supported by a grant from the U.S.Israeli Binational Science Foundation. Work on this paper by Pankaj K. Agarwal has also been supported by NSF Grant CCR9301259, an NYI award, and matching funds from Xerox Corporation. Work on this paper by Boris Aronov has also been supported by NSF Grant CCR9211541 and by a Sloan Research Fellowship. Work on this paper by J'anos Pach, Richard Pollack, and Micha Sharir has been supported by NSF Grants CCR9122103 and CCR9424398. Work by J'anos Pach was also supp...
Constructing Piecewise Linear Homeomorphisms
, 1994
"... Let P = fp 1 ; : : : ; p n g and Q = fq 1 ; : : : ; q n g be two point sets lying in the interior of rectangles in the plane. We show how to construct a piecewise linear homeomorphism of size O(n 2 ) between the rectangles which maps p i to q i for each i. This bound is optimal in the worst case; ..."
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Cited by 32 (7 self)
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Let P = fp 1 ; : : : ; p n g and Q = fq 1 ; : : : ; q n g be two point sets lying in the interior of rectangles in the plane. We show how to construct a piecewise linear homeomorphism of size O(n 2 ) between the rectangles which maps p i to q i for each i. This bound is optimal in the worst case; i.e., there exist point sets for which any piecewise linear homeomorphism has size\Omega\Gamma n 2 ).
Extremal Problems for Geometric Hypergraphs
 Discrete Comput. Geom
, 1998
"... A geometric hypergraph H is a collection of idimensional simplices, called hyperedges or, simply, edges, induced by some (i + 1)tuples of a vertex set V in general position in dspace. The topological structure of geometric graphs, i.e., the case d = 2; i = 1, has been studied extensively, and it ..."
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Cited by 24 (2 self)
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A geometric hypergraph H is a collection of idimensional simplices, called hyperedges or, simply, edges, induced by some (i + 1)tuples of a vertex set V in general position in dspace. The topological structure of geometric graphs, i.e., the case d = 2; i = 1, has been studied extensively, and it proved to be instrumental for the solution of a wide range of problems in combinatorial and computational geometry. They include the kset problem, proximity questions, bounding the number of incidences between points and lines, designing various efficient graph drawing algorithms, etc. In this paper, we make an attempt to generalize some of these tools to higher dimensions. We will mainly consider extremal problems of the following type. What is the largest number of edges (isimplices) that a geometric hypergraph of n vertices can have without containing certain forbidden configurations? In particular, we discuss the special cases when the forbidden configurations are k intersecting edges...
Generalized DavenportSchinzel sequences: results, problems, and applications
, 1994
"... We survey in detail... ..."
Experiments on the Minimum Linear Arrangement Problem
 Sistemes Informàtics, 2001. (Preliminary version in Alex ’98 — Building Bridges between Theory and Applications
, 2001
"... This paper deals with the Minimum Linear Arrangement problem from an experimental point of view. Using a testsuite of sparse graphs, we experimentally compare several algorithms to obtain upper and lower bounds for this problem. The algorithms considered include Successive Augmentation heuristics, ..."
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Cited by 14 (0 self)
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This paper deals with the Minimum Linear Arrangement problem from an experimental point of view. Using a testsuite of sparse graphs, we experimentally compare several algorithms to obtain upper and lower bounds for this problem. The algorithms considered include Successive Augmentation heuristics, Local Search heuristics and Spectral Sequencing. The testsuite is based on two random models and "real life" graphs. As a consequence of this study, two main conclusions can be drawn: On one hand, the best approximations are usually obtained using Simulated Annealing, which involves a large amount of computation time. However, solutions found with Spectral Sequencing are close to the ones found with Simulated Annealing and can be obtained in significantly less time. On the other hand, we notice that there exists a big gap between the best obtained upper bounds and the best obtained lower bounds. These two facts together show that, in practice, finding lower and upper bounds for the Minimum ...
Crossing Numbers: Bounds and Applications
 I. B'AR'ANY AND K. BOROCZKY, BOLYAI SOCIETY MATHEMATICAL STUDIES 6
, 1997
"... We give a survey of techniques for deriving lower bounds and algorithms for constructing upper bounds for several variations of the crossing number problem. Our aim is to emphasize the more general results or those results which have an algorithmic flavor, including the recent results of the autho ..."
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Cited by 13 (5 self)
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We give a survey of techniques for deriving lower bounds and algorithms for constructing upper bounds for several variations of the crossing number problem. Our aim is to emphasize the more general results or those results which have an algorithmic flavor, including the recent results of the authors. We also show applications of crossing numbers to other areas of discrete mathematics, like discrete geometry.
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 13 (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.
Crossing numbers of Sierpińskilike graphs
 J. Graph Theory
, 2005
"... The crossing number of Sierpiński graphs S(n, k) and their regularizations S + (n, k) and S ++ (n, k) is studied. Explicit drawings of these graphs are presented and proved to be optimal for S + (n, k) and S ++ (n, k) for every n ≥ 1 and k ≥ 1. These are the first nontrivial families of graphs of “f ..."
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Cited by 10 (0 self)
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The crossing number of Sierpiński graphs S(n, k) and their regularizations S + (n, k) and S ++ (n, k) is studied. Explicit drawings of these graphs are presented and proved to be optimal for S + (n, k) and S ++ (n, k) for every n ≥ 1 and k ≥ 1. These are the first nontrivial families of graphs of “fractal ” type whose crossing number is known.
The minor crossing number of graphs with an excluded minor
 ELECTRONIC J. COMBINATORICS
, 2008
"... The minor crossing number of a graph G is the minimum crossing number of a graph that contains G as a minor. It is proved that for every graph H there is a constant c, such that every graph G with no Hminor has minor crossing number at most cV (G). ..."
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Cited by 7 (2 self)
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The minor crossing number of a graph G is the minimum crossing number of a graph that contains G as a minor. It is proved that for every graph H there is a constant c, such that every graph G with no Hminor has minor crossing number at most cV (G).
Untangling a Polygon
"... The following problem was raised by M. Watanabe. Let P be a selfintersecting closed polygon with n vertices in general position. How manys steps does it take to untangle P , i.e., to turn it into a simple polygon, if in each step we can arbitrarily relocate one of its vertices. It is shown that in ..."
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Cited by 6 (0 self)
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The following problem was raised by M. Watanabe. Let P be a selfintersecting closed polygon with n vertices in general position. How manys steps does it take to untangle P , i.e., to turn it into a simple polygon, if in each step we can arbitrarily relocate one of its vertices. It is shown that in some cases one has to move all but at most O((n log n) 2=3 ) vertices. On the other hand, every polygon P can be untangled in at most n p n) steps. Some related questions are also considered. 1