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21
Applications of the crossing number
 Proc. 10th Annual ACM Symp. on Computational Geometry
, 1994
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Counting patternfree set partitions. II: Noncrossing and other hypergraphs
 J. Combin
, 2000
"... A (multi)hypergraph H with vertices in N contains a permutation p = a 1 a 2: : : a k of 1; 2; : : : ; k if one can reduce H by omitting vertices from the edges so that the resulting hypergraph is isomorphic, via an increasing mapping, to H p = (fi; k + a i g: i = 1; : : : ; k). We formulate six co ..."
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Cited by 20 (9 self)
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A (multi)hypergraph H with vertices in N contains a permutation p = a 1 a 2: : : a k of 1; 2; : : : ; k if one can reduce H by omitting vertices from the edges so that the resulting hypergraph is isomorphic, via an increasing mapping, to H p = (fi; k + a i g: i = 1; : : : ; k). We formulate six conjectures stating that if H has n vertices and does not contain p then the size of H is O(n) and the number of such Hs is O(c n). The latter part generalizes the StanleyWilf conjecture on permutations. Using generalized DavenportSchinzel sequences, we prove the conjectures with weaker bounds O(nfi(n)) and O(fi(n) n), where fi(n) ! 1 very slowly. We prove the conjectures fully if p first increases and then decreases or if p
Generalized DavenportSchinzel sequences: results, problems, and applications
, 1994
"... We survey in detail... ..."
Notes on geometric graph theory
 Discrete and Computational Geometry: Papers from DIMACS special year, volume 6 of DIMACS series, 273–285, AMS
, 1991
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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 9 (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 k&sup3;(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.
Geometric Graphs With No SelfIntersecting Path Of Length Three
"... Let G be a geometric graph with n vertices, i.e., a graph drawn in the plane with straightline edges. It is shown that if G has no selfintersecting path of length 3, then its number of edges is O(n log n). This result is asymptotically tight. Analogous questions for curvilinear drawings and for ..."
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Cited by 8 (5 self)
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Let G be a geometric graph with n vertices, i.e., a graph drawn in the plane with straightline edges. It is shown that if G has no selfintersecting path of length 3, then its number of edges is O(n log n). This result is asymptotically tight. Analogous questions for curvilinear drawings and for longer paths are also considered.
Notes on Large Angle Crossing Graphs
, 2010
"... A graph G is an α angle crossing (αAC) graph if every pair of crossing edges in G intersect at an angle of at least α. The concept of right angle crossing (RAC) graphs (α = π/2) was recently introduced by Didimo et al. [6]. It was shown that any RAC graph with n vertices has at most 4n −10 edges and ..."
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Cited by 6 (0 self)
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A graph G is an α angle crossing (αAC) graph if every pair of crossing edges in G intersect at an angle of at least α. The concept of right angle crossing (RAC) graphs (α = π/2) was recently introduced by Didimo et al. [6]. It was shown that any RAC graph with n vertices has at most 4n −10 edges and that there are infinitely many values of n for which there exists a RAC graph with n vertices and 4n − 10 edges. In this paper, we give upper and lower bounds for the number of edges in αAC graphs for all 0 < α < π/2. 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...
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