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43
Which crossing number is it, anyway
 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
, 1998
"... A drawing of a graph G is a mapping which assigns to each vertex a point of the plane and to each edge a simple continuous arc connecting the corresponding two points. The crossing number of G is the minimum number of crossing points in any drawing of G. We define two new parameters, as follows. The ..."
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Cited by 45 (8 self)
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A drawing of a graph G is a mapping which assigns to each vertex a point of the plane and to each edge a simple continuous arc connecting the corresponding two points. The crossing number of G is the minimum number of crossing points in any drawing of G. We define two new parameters, as follows. The pairwise crossing number (resp. the oddcrossing number) of G is the minimum number of pairs of edges that cross (resp. cross an odd number of times) over all drawings of G. We prove that the largest of these numbers (the crossing number) cannot exceed twice the square of the smallest (the oddcrossing number). Our proof is based on the following generalization of an old result of Hanani, which is of independent interest. Let G be a graph and let E0 be a subset of its edges such that there is a drawing of G, in which every edge belonging to E0 crosses any other edge an even number of times. Then G can be redrawn so that the elements of E0 are not involved in any crossing. Finally, we show that the determination of each of these parameters is an NPhard problem and it is NPcomplete in the case of the crossing number and the oddcrossing number. 1
Lenses in arrangements of pseudocircles and their applications
 J. ACM
, 2004
"... Abstract. A collection of simple closed Jordan curves in the plane is called a family of pseudocircles if any two of its members intersect at most twice. A closed curve composed of two subarcs of distinct Work on this article by P. Agarwal, J. Pach, and M. Sharir has been supported by a joint grant ..."
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Cited by 24 (10 self)
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Abstract. A collection of simple closed Jordan curves in the plane is called a family of pseudocircles if any two of its members intersect at most twice. A closed curve composed of two subarcs of distinct Work on this article by P. Agarwal, J. Pach, and M. Sharir has been supported by a joint grant from the U.S.–Israel Binational Science Foundation. Work by P. Agarwal has also been supported by NSF grants EIA9870724, EIA9972879, ITR333
The Complexity of Planarity Testing
, 2000
"... We clarify the computational complexity of planarity testing, by showing that planarity testing is hard for L, and lies in SL. This nearly settles the question, since it is widely conjectured that L = SL [25]. The upper bound of SL matches the lower bound of L in the context of (nonuniform) circ ..."
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Cited by 23 (7 self)
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We clarify the computational complexity of planarity testing, by showing that planarity testing is hard for L, and lies in SL. This nearly settles the question, since it is widely conjectured that L = SL [25]. The upper bound of SL matches the lower bound of L in the context of (nonuniform) circuit complexity, since L/poly is equal to SL/poly. Similarly, we show that a planar embedding, when one exists, can be found in FL SL . Previously, these problems were known to reside in the complexity class AC 1 , via a O(log n) time CRCW PRAM algorithm [22], although planarity checking for degreethree graphs had been shown to be in SL [23, 20].
New lower bounds for the number of (≤ k)edges and the rectilinear crossing number of Kn, Discrete Comput
 Geom
"... We provide a new lower bound on the number of ( ≤ k)edges of a set of n points in the plane in general position. We show that for 0 ≤ k ≤ ⌊ n−2 2 ⌋ the number of ( ≤ k)edges is at least ( ) k ∑ k + 2 Ek(S) ≥ 3 + (3j − n + 3), 2 j= ⌊ n 3 ⌋ which, for k ≥ ⌊ n 3 ⌋, improves the previous best lower ..."
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Cited by 18 (3 self)
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We provide a new lower bound on the number of ( ≤ k)edges of a set of n points in the plane in general position. We show that for 0 ≤ k ≤ ⌊ n−2 2 ⌋ the number of ( ≤ k)edges is at least ( ) k ∑ k + 2 Ek(S) ≥ 3 + (3j − n + 3), 2 j= ⌊ n 3 ⌋ which, for k ≥ ⌊ n 3 ⌋, improves the previous best lower bound in [7]. As a main consequence, we obtain a new lower bound on the rectilinear crossing number of the complete graph or, in other words, on the minimum number of convex quadrilaterals determined by n points in the plane in general position. We show that the crossing number is at least
Removing even crossings
 J. COMBINAT. THEORY, SER. B
, 2005
"... An edge in a drawing of a graph is called even if it intersects every other edge of the graph an even number of times. Pach and Tóth proved that a graph can always be redrawn such that its even edges are not involved in any intersections. We give a new, and significantly simpler, proof of a slightly ..."
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Cited by 15 (8 self)
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An edge in a drawing of a graph is called even if it intersects every other edge of the graph an even number of times. Pach and Tóth proved that a graph can always be redrawn such that its even edges are not involved in any intersections. We give a new, and significantly simpler, proof of a slightly stronger statement. We show two applications of this strengthened result: an easy proof of a theorem of Hanani and Tutte (the only proof we know of not to use Kuratowski’s theorem), and the result that the odd crossing number of a graph equals the crossing number of the graph for values of at most 3. The paper begins with a disarmingly simple proof of a weak (but standard) version of the theorem by Hanani and Tutte.
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.
Genetic algorithms for drawing bipartite graphs
 INTERN. J. COMPUTER MATH
, 1994
"... This paper introduces genetic algorithms for the level permutation problem (LPP). The problem is to minimize the number of edge crossings in a bipartite graph when the order of vertices in one of the two vertex subsets is fixed. We show that genetic algorithms outperform the previously known heurist ..."
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Cited by 13 (3 self)
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This paper introduces genetic algorithms for the level permutation problem (LPP). The problem is to minimize the number of edge crossings in a bipartite graph when the order of vertices in one of the two vertex subsets is fixed. We show that genetic algorithms outperform the previously known heuristics especially when applied to low density graphs. Values for various parameters of genetic LPP algorithms are tested.
CrossingNumber Critical Graphs have Bounded Pathwidth
, 2000
"... . The crossing number of a graph G, denoted by cr(G), is dened as the smallest possible number of edgecrossings in a drawing of G in the plane. A graph G is crossingcritical if cr(G e) < cr(G) for all edges e of G. We prove that crossingcritical graphs have \bounded pathwidth" (by a function ..."
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Cited by 13 (1 self)
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. The crossing number of a graph G, denoted by cr(G), is dened as the smallest possible number of edgecrossings in a drawing of G in the plane. A graph G is crossingcritical if cr(G e) < cr(G) for all edges e of G. We prove that crossingcritical graphs have \bounded pathwidth" (by a function of the crossing number), which roughly means that such graphs are made up of small pieces joined in a linear way on small cutsets. Equivalently, a crossingcritical graph cannot contain a subdivision of a \large" binary tree. This assertion was conjectured earlier by Salazar in [J. Geelen, B. Richter, G. Salazar, Embedding grids on surfaces, manuscript, 2000]. 1 Introduction We begin with the most important denitions here. Additional denitions and comments will be presented in the subsequent section. If % : [0; 1] ! IR 2 is a simple continuous function, then %([0; 1]) is a simple curve, and %((0; 1)) is a simple open curve. Denition. A graph G is drawn in the plane if the ver...
On conway’s thrackle conjecture
 Proc. 11th ACM Symp. on Computational Geometry
, 1995
"... A thrackle is a graph drawn in the plane so that its edges are represented by Jordan arcs and any two distinct arcs either meet at exactly one common vertex or cross at exactly one point interior to both arcs. About forty years ago, J. H. Conway conjectured that the number of edges of a thrackle can ..."
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Cited by 12 (2 self)
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A thrackle is a graph drawn in the plane so that its edges are represented by Jordan arcs and any two distinct arcs either meet at exactly one common vertex or cross at exactly one point interior to both arcs. About forty years ago, J. H. Conway conjectured that the number of edges of a thrackle cannot exceed the number of its vertices. We show that a thrackle has at most twice as many edges as vertices. Some related problems and generalizations are also considered. 1