Abstract
A topological graph is k-quasi-planar if it does not contain k pairwise crossing edges. A topological graph is simple if every pair of its edges intersect at most once (either at a vertex or at their intersection). In 1996, Pach, Shahrokhi, and Szegedy [16] showed that every n-vertex simple k-quasi-planar graph contains at most \(O\left(n(\log n)^{2k-4}\right)\) edges. This upper bound was recently improved (for large k) by Fox and Pach [8] to n(logn)O(logk). In this note, we show that all such graphs contain at most \((n\log^2n )2^{\alpha^{c_k}(n)}\) edges, where α(n) denotes the inverse Ackermann function and c k is a constant that depends only on k.
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Suk, A. (2012). k-Quasi-Planar Graphs. In: van Kreveld, M., Speckmann, B. (eds) Graph Drawing. GD 2011. Lecture Notes in Computer Science, vol 7034. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25878-7_26
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DOI: https://doi.org/10.1007/978-3-642-25878-7_26
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