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Applications of the crossing number
 In Proc. 10th Annu. ACM Sympos. Comput. Geom
, 1994
"... Abstract. The crossing number of a graph G is the minimum number of crossings in a drawing of G. The determination of the crossing number is an NPcomplete problem. We present two general lower bounds for the crossing number, and survey their applications and generalizations. 1 ..."
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Cited by 28 (6 self)
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Abstract. The crossing number of a graph G is the minimum number of crossings in a drawing of G. The determination of the crossing number is an NPcomplete problem. We present two general lower bounds for the crossing number, and survey their applications and generalizations. 1
On Geometric Graphs With No K Pairwise Parallel Edges
 Discrete Comput. Geom
, 1997
"... A geometric graph is a graph G = (V; E) drawn in the plane so that the vertex set V consists of points in general position and the edge set E consists of straight line segments between points of V . Two edges of a geometric graph are said to be parallel , if they are opposite sides of a convex quadr ..."
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Cited by 19 (1 self)
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A geometric graph is a graph G = (V; E) drawn in the plane so that the vertex set V consists of points in general position and the edge set E consists of straight line segments between points of V . Two edges of a geometric graph are said to be parallel , if they are opposite sides of a convex quadrilateral. In this paper we show that, for any fixed k 3, any geometric graph on n vertices with no k pairwise parallel edges contains at most O(n) edges, and any geometric graph on n vertices with no k pairwise crossing edges contains at most O(n log n) edges. We also prove a conjecture of Kupitz that any geometric graph on n vertices with no pair of parallel edges contains at most 2n \Gamma 2 edges. 1 Introduction A geometric graph is a graph G = (V; E) drawn in the plane so that the vertex set V consists of points in general position and the edge set E consists of straight line segments between points of V . See [9] for a survey of results about geometric graphs. Two edges of a geomet...
Note on Geometric Graphs
, 1999
"... A geometric graph is a graph drawn in the plane so that the vertices are represented by points in general position and the edges are represented by straight line segments connecting the corresponding points. We show that a geometric graph of n vertices with no k+1 pairwise disjoint edges has at most ..."
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Cited by 13 (3 self)
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A geometric graph is a graph drawn in the plane so that the vertices are represented by points in general position and the edges are represented by straight line segments connecting the corresponding points. We show that a geometric graph of n vertices with no k+1 pairwise disjoint edges has at most 2 9 k 2 n edges.
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.
TRACES OF FINITE SETS: EXTREMAL PROBLEMS AND GEOMETRIC APPLICATIONS
, 1992
"... Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as HS = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VCdimension) of H is the size of the largest subset S for which HS has 2 S edges. Hypergraphs of small VCdimension play a central role in many areas o ..."
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Cited by 13 (0 self)
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Given a hypergraph H and a subset S of its vertices, the trace of H on S is defined as HS = {E ∩ S: E ∈ H}. The Vapnik–Chervonenkis dimension (VCdimension) of H is the size of the largest subset S for which HS has 2 S edges. Hypergraphs of small VCdimension play a central role in many areas of statistics, discrete and computational geometry, and learning theory. We survey some of the most important results related to this concept with special emphasis on (a) hypergraph theoretic methods and (b) geometric applications.
Geometric Graphs with Few Disjoint Edges
, 1998
"... A geometric graph is a graph drawn in the plane so that the vertices are represented by points in general position, the edges are represented by straight line segments connecting the corresponding points. Improving a result of Pach and Töröcsik, we show that a geometric graph on n vertices with no k ..."
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Cited by 10 (2 self)
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A geometric graph is a graph drawn in the plane so that the vertices are represented by points in general position, the edges are represented by straight line segments connecting the corresponding points. Improving a result of Pach and Töröcsik, we show that a geometric graph on n vertices with no k + 1 pairwise disjoint edges has at most k³(n + 1) edges. On the other hand, we construct geometric graphs with n vertices and approximately 3/2 (k  1)n edges, containing no k + 1 pairwise disjoint edges. We also improve both the lower and upper bounds of Goddard, Katchalski and Kleitman on the maximum number of edges in a geometric graph with no four pairwise disjoint edges.
Relaxing planarity for topological graphs
 Discrete and Computational Geometry, Lecture Notes in Comput. Sci., 2866
, 2003
"... Abstract. According to Euler’s formula, every planar graph with n vertices has at most O(n) edges. How much can we relax the condition of planarity without violating the conclusion? After surveying some classical and recent results of this kind, we prove that every graph of n vertices, which can be ..."
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Cited by 6 (3 self)
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Abstract. According to Euler’s formula, every planar graph with n vertices has at most O(n) edges. How much can we relax the condition of planarity without violating the conclusion? After surveying some classical and recent results of this kind, we prove that every graph of n vertices, which can be drawn in the plane without three pairwise crossing edges, has at most O(n) edges. For straightline drawings, this statement has been established by Agarwal et al., using a more complicated argument, but for the general case previously no bound better than O(n 3/2) was known. 1
On Crossing Sets, Disjoint Sets and the Pagenumber
, 1998
"... Let G = (V; E) be a tpartite graph with n vertices and m edges, where the partite sets are given. We present an O(n 2 m 1:5 ) time algorithm to construct drawings of G in the plane so that the size of the largest set of pairwise crossing edges, and at the same time, the size of the largest set ..."
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Cited by 6 (0 self)
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Let G = (V; E) be a tpartite graph with n vertices and m edges, where the partite sets are given. We present an O(n 2 m 1:5 ) time algorithm to construct drawings of G in the plane so that the size of the largest set of pairwise crossing edges, and at the same time, the size of the largest set of disjoint (pairwise noncrossing) edges are O( p t \Delta m). As an application we embed G in a book of O( p t \Delta m) pages, in O(n 2 m 1:5 ) time, resolving an open question for the pagenumber problem. A similar result is obtained for the dual of the pagenumber or the queuenumber. Our algorithms are obtained by derandomizing a probabilistic proof. 1 Introduction and Summary 1.1 Preliminaries Throughout this paper G = (V; E) is an undirected graph with jV j = n and jEj = m. A linear ordering of a set S is a bijection from S to f1; 2; : : : ; jSjg. Let h be a linear ordering of V . Consider a drawing of G that is obtained by placing the vertices along a straight line in the pl...
On the chromatic number of some geometric type Kneser graphs
, 2004
"... We estimate the chromatic number of graphs whose vertex set is the set of edges of a complete geometric graph on n points, and adjacency is defined in terms of geometric disjointness or geometric intersection. ..."
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Cited by 5 (4 self)
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We estimate the chromatic number of graphs whose vertex set is the set of edges of a complete geometric graph on n points, and adjacency is defined in terms of geometric disjointness or geometric intersection.