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26
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
Planarizing Graphs  A Survey and Annotated Bibliography
, 1999
"... Given a finite, undirected, simple graph G, we are concerned with operations on G that transform it into a planar graph. We give a survey of results about such operations and related graph parameters. While there are many algorithmic results about planarization through edge deletion, the results abo ..."
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Cited by 33 (0 self)
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Given a finite, undirected, simple graph G, we are concerned with operations on G that transform it into a planar graph. We give a survey of results about such operations and related graph parameters. While there are many algorithmic results about planarization through edge deletion, the results about vertex splitting, thickness, and crossing number are mostly of a structural nature. We also include a brief section on vertex deletion. We do not consider parallel algorithms, nor do we deal with online algorithms.
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
Toward the Rectilinear Crossing Number of K_n : New Embeddings, Upper Bounds, and Asymptotics
, 2000
"... Scheinerman and Wilf [SW94] assert that "an important open problem in the study of graph embeddings is to determine the rectilinear crossing number of the complete graph Kn ." A rectilinear embedding or drawing of Kn is an arrangement of n vertices in the plane, every pair of which is connected by ..."
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Cited by 22 (1 self)
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Scheinerman and Wilf [SW94] assert that "an important open problem in the study of graph embeddings is to determine the rectilinear crossing number of the complete graph Kn ." A rectilinear embedding or drawing of Kn is an arrangement of n vertices in the plane, every pair of which is connected by an edge that is a line segment. We assume that no three vertices are collinear. The rectilinear crossing number of Kn is the fewest number of edge crossings attainable over all planar rectilinear embeddings of Kn . For each n we construct a rectilinear embedding of Kn that has the fewest number of edge crossings and the best asymptotics known to date. Moreover, we give some alternative infinite families of embeddings of Kn with good asymptotics. Finally, we mention some old and new open problems.
Thirteen Problems on Crossing Numbers
, 2000
"... The crossing number of a graph G is the minimum number of crossings in a drawing of G. We introduce several variants of this definition, and present a list of related open problems. The first item is Zarankiewicz's classical conjecture about crossing numbers of complete bipartite graphs, the la ..."
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Cited by 15 (0 self)
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The crossing number of a graph G is the minimum number of crossings in a drawing of G. We introduce several variants of this definition, and present a list of related open problems. The first item is Zarankiewicz's classical conjecture about crossing numbers of complete bipartite graphs, the last ones are new and less carefully tested. In Section 5, we state some conjectures about the expected values of various crossing numbers of random graphs, and prove a large deviation result.
Planar decompositions and the crossing number of graphs with an excluded minor
 IN GRAPH DRAWING 2006; LECTURE NOTES IN COMPUTER SCIENCE 4372
, 2007
"... Tree decompositions of graphs are of fundamental importance in structural and algorithmic graph theory. Planar decompositions generalise tree decompositions by allowing an arbitrary planar graph to index the decomposition. We prove that every graph that excludes a fixed graph as a minor has a planar ..."
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Cited by 14 (1 self)
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Tree decompositions of graphs are of fundamental importance in structural and algorithmic graph theory. Planar decompositions generalise tree decompositions by allowing an arbitrary planar graph to index the decomposition. We prove that every graph that excludes a fixed graph as a minor has a planar decomposition with bounded width and a linear number of bags. The crossing number of a graph is the minimum number of crossings in a drawing of the graph in the plane. We prove that planar decompositions are intimately related to the crossing number. In particular, a graph with bounded degree has linear crossing number if and only if it has a planar decomposition with bounded width and linear order. It follows from the above result about planar decompositions that every graph with bounded degree and an excluded minor has linear crossing number. Analogous results are proved for the convex and rectilinear crossing numbers. In particular, every graph with bounded degree and bounded treewidth has linear convex crossing number, and every K3,3minorfree graph with bounded degree has linear rectilinear crossing number.
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.
The Rectilinear Crossing Number of K_10 is 62
 Electron. J. Combin., 8(1):Research Paper
, 2000
"... The rectilinear crossing number of a graph G is the minimum number of edge crossings that can occur in any drawing of G in which the edges are straight line segments and no three vertices are collinear. This number has been known for G = K n if n # 9. Using a combinatorial argument we show that fo ..."
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Cited by 11 (0 self)
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The rectilinear crossing number of a graph G is the minimum number of edge crossings that can occur in any drawing of G in which the edges are straight line segments and no three vertices are collinear. This number has been known for G = K n if n # 9. Using a combinatorial argument we show that for n =10the number is 62.