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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 42 (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
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 26 (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
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 14 (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.
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 11 (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
A Proposed Algorithm for Calculating the Minimum Crossing Number of a Graph
 Western Michigan University
, 1995
"... In this paper we present a branchandbound algorithm for finding the minimum crossing number of a graph. We begin with the vertex set and add edges by selecting every legal option for creating a crossing or not. After each edge is added we determine if the resulting partial graph is planar. We cont ..."
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Cited by 5 (0 self)
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In this paper we present a branchandbound algorithm for finding the minimum crossing number of a graph. We begin with the vertex set and add edges by selecting every legal option for creating a crossing or not. After each edge is added we determine if the resulting partial graph is planar. We continue adding edges until either all edges have been added or we reach a point where the graph cannot be completed as started. At this point we backtrack to see if the graph can be drawn with fewer crossings by selecting other options when adding edges. keywords: Crossing Number, Algorithm 1 Introduction Determining the crossing number of a graph is an important problem with applications in areas such as circuit design and network configuration [17]. It is this importance that has driven our work in finding the minimum crossing number of a graph. Informally, the crossing number of a graph G, denoted (G), is the minimum number of crossings among all good drawings of G in the plane, where a g...
Unavoidable Configurations in Complete Topological Graphs
 Proc. Graph Drawing 2000., LNCS
, 1984
"... A topological graph is a graph drawn in the plane so that its vertices are represented by points, and its edges are represented by Jordan curves connecting the corresponding points, with the property that any two curves have at most one point in common. We define two canonical classes of topological ..."
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Cited by 5 (3 self)
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A topological graph is a graph drawn in the plane so that its vertices are represented by points, and its edges are represented by Jordan curves connecting the corresponding points, with the property that any two curves have at most one point in common. We define two canonical classes of topological complete graphs, and prove that every topological complete graph with n vertices has a canonical subgraph of size at least c log 1/8 n, which belongs to one of these classes. We also show that every complete topological graph with n vertices has a noncrossing subgraph isomorphic to any fixed tree with at most c log 1/6 n vertices.
Approximating the Crossing Number of Graphs Embeddable In Any Orientable Surface
"... The crossing number of a graph is the least number of pairwise edge crossings in a drawing of the graph in the plane. We provide an O(n log n) time constant factor approximation algorithm for the crossing number of a graph of bounded maximum degree which is “densely enough” embeddable in an arbitrar ..."
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Cited by 5 (3 self)
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The crossing number of a graph is the least number of pairwise edge crossings in a drawing of the graph in the plane. We provide an O(n log n) time constant factor approximation algorithm for the crossing number of a graph of bounded maximum degree which is “densely enough” embeddable in an arbitrary fixed orientable surface. Our approach combines some known tools with a powerful new lower bound on the crossing number of an embedded graph. This result extends previous results that gave such approximations in particular cases of projective, toroidal or apex graphs; it is a qualitative improvement over previously published algorithms that constructed lowcrossingnumber drawings of embeddable graphs without giving any approximation guarantees. No constant factor approximation algorithms for the crossing number problem over comparably rich classes of graphs are known to date.
Parallel Computation of the Minimum Crossing Number of a Graph
 Proc. of the 8 th SIAM Conf. on Parallel Process. for Sci. Comput
, 1997
"... Determining the crossing number of a graph is an important problem with applications in areas such as circuit design and network configuration. In this paper we present the first parallel algorithm for solving this combinatorial optimization problem. This branchandbound algorithm, which adds and d ..."
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Cited by 4 (1 self)
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Determining the crossing number of a graph is an important problem with applications in areas such as circuit design and network configuration. In this paper we present the first parallel algorithm for solving this combinatorial optimization problem. This branchandbound algorithm, which adds and deletes crossings in an organized fashion, presents us with the opportunity to verify many conjectures which are decades old, as well as pursuing future work in efficient circuit design. 1 Introduction Optimal circuit layout and network design are two problems that are becoming more important as the number of computers grows rapidly and their capabilities increase. Unfortunately the applications are obvious but the theory available to help solve the problem is quite deficient. Determining the crossing number of a graph is a problem whose solution could improve the state of the theory in this important application area. It is this importance that has driven our work in finding the minimum cro...