Randomized rounding: a technique for provably good algorithms and algorithmic proofs
Combinatorica - Theory of Computing
On VLSI layouts of the star graph and related networks
Integration, the VLSI Journal
The book crossing number of a graph
Journal of Graph Theory
Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms
Journal of the ACM (JACM)
Which crossing number is it anyway?
Journal of Combinatorial Theory Series B
Improved bounds for the unsplittable flow problem
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Improved Approximations of Crossings in Graph Drawings and VLSI Layout Areas
SIAM Journal on Computing
Crossing numbers of random graphs
Random Structures & Algorithms - Special issue: Proceedings of the tenth international conference "Random structures and algorithms"
Expander flows, geometric embeddings and graph partitioning
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Bounds for convex crossing numbers
COCOON'03 Proceedings of the 9th annual international conference on Computing and combinatorics
Disjoint edges in topological graphs
IJCCGGT'03 Proceedings of the 2003 Indonesia-Japan joint conference on Combinatorial Geometry and Graph Theory
Coloring kk-free intersection graphs of geometric objects in the plane
Proceedings of the twenty-fourth annual symposium on Computational geometry
A Separator Theorem for String Graphs and Its Applications
WALCOM '09 Proceedings of the 3rd International Workshop on Algorithms and Computation
On grids in topological graphs
Proceedings of the twenty-fifth annual symposium on Computational geometry
A bipartite strengthening of the Crossing Lemma
Journal of Combinatorial Theory Series B
A bipartite strengthening of the crossing lemma
GD'07 Proceedings of the 15th international conference on Graph drawing
A separator theorem for string graphs and its applications
Combinatorics, Probability and Computing
Odd crossing number is not crossing number
GD'05 Proceedings of the 13th international conference on Graph Drawing
On graph crossing number and edge planarization
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Coloring Kk-free intersection graphs of geometric objects in the plane
European Journal of Combinatorics
Disjoint edges in topological graphs
IJCCGGT'03 Proceedings of the 2003 Indonesia-Japan joint conference on Combinatorial Geometry and Graph Theory
String graphs and incomparability graphs
Proceedings of the twenty-eighth annual symposium on Computational geometry
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The crossing number cr(G) of a graph G is the minimum possible number of edge crossings in a drawing of G in the plane, while the pair-crossing number pcr(G) is the smallest number of pairs of edges that cross in a drawing of G in the plane. While cr(G) ≥ pcr(G) holds trivially, it is not known whether a strict inequality can ever occur (this question was raised by Mohar and Pach and Tóth). We aim at bounding cr(G) in terms of pcr(G). Using the methods of Leighton and Rao, Bhatt and Leighton, and Even, Guha and Schieber, we prove that cr(G) = O(log3 n(pcr(G) + ssqd(G))), where n = |V(G)| and ssqd(G) = Σv∈V(G) degG(v)2. One of the main steps is an analogy of the well-known lower bound cr(G) = Ω(b(G)2) - O(ssqd(G)), where b(G) is the bisection width of G, that is, the smallest number of edges that have to be removed so that no component of the resulting graph has more than 2/3 n vertices. We show that pcr(G) = Ω(b(G)2/log2 n) - O(ssqd(G)). We also prove by similar methods that a graph G with crossing number k = cr(G) C√ssqd(G) m log2 n has a nonplanar subgraph on at most O(Δnm log2 n/k) vertices, where m is the number of edges, Δ is the maximum degree in G, and C is a suitable sufficiently large constant.