Abstract
The k-planar crossing number of a graph is the minimum number of crossings of its edges over all possible drawings of the graph in k planes. We propose algorithms and methods for k-planar drawings of general graphs together with lower bound techniques. We give exact results for the k-planar crossing number of K 2 k + 1, q, for k ≥ 2. We prove tight bounds for complete graphs.
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Aggarwal, A., Klawe, M., Shor, P.: Multi-layer grid embeddings for VLSI. Algorithmica 6, 129–151 (1991)
Beineke, L.W.: Complete bipartite graphs: Decomposition into planar subgraphs. In: Harary, F. (ed.) A Seminar on Graph Theory, Selected Topics in Mathematics. ch. 7, pp. 43–53. Holt, Rinehart and Winston (1967)
Beineke, L.W.: Biplanar graphs: A survey. Computers and Mathematics with Applications 34, 1–8 (1997)
Beineke, L.W., Harary, F., Moon, J.W.: On the thickness of the complete bipartite graphs. In: Proc. of the Cambridge Philosophical Society, vol. 60, pp. 1–5 (1964)
Czabarka, É., Sýkora, O., Székely, L.A., Vrťo, I.: Biplanar crossing numbers: A survey of results and problems. In: Fleiner, T., Katona, G.O.H. (eds.) Finite and Infinite Combinatorics, Akadémia Kiadó, Budapest. Bolyai Society Mathematical Studies (to appear)
Kleitman, D.J.: The crossing number of K.,n. J. Combinatorial Theory 9, 315–323 (1970)
Leighton, T.F.: Complexity Issues in VLSI. MIT Press, Cambridge (1983)
Nash-Williams, J.A.: Edge disjoint spanning trees of finite graphs. J. London Math. Soc. 36, 445–450 (1961)
Owens, A.: On the biplanar crossing number. IEEE Transactions on Circuit Theory 18, 277–280 (1971)
Richter, R.B., Širáň, J.: The crossing number of K.,n in a surface. J. Graph Theory 21, 51–54 (1996)
Shahrokhi, F., Sýkora, O., Székely, L.A., Vrťo, I.: The book crossing number of graphs. J. Graph Theory 21, 413–424 (1996)
Sýkora, O., Székely, L.A., Vrťo, I.: Crossing numbers and biplanar crossing numbers: using the probabilistic method (submitted)
Shahrokhi, F., Sýkora, O., Székely, L.A., Vrťo, I.: Bounds for convex crossing numbers. In: Warnow, T.J., Zhu, B. (eds.) COCOON 2003. LNCS, vol. 2697, pp. 487–495. Springer, Heidelberg (2003)
Truszczyński, M.: Decomposition of graphs into forests with bounded maximum degree. Discrete Mathematics 98, 207–222 (1991)
White, A.T., Beineke, L.W.: Topological graph theory. In: Beineke, L.W., Wilson, R.J. (eds.) Selected Topics in Graph Theory, pp. 15–50. Academic Press, New York (1978)
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Shahrokhi, F., Sýkora, O., Székely, L.A., Vrt’o, I. (2004). Bounds and Methods for k-Planar Crossing Numbers. In: Liotta, G. (eds) Graph Drawing. GD 2003. Lecture Notes in Computer Science, vol 2912. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24595-7_4
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DOI: https://doi.org/10.1007/978-3-540-24595-7_4
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