Abstract
We give drawings of a complete graphK n withO(n 4 log2 g/g) many crossings on an orientable or nonorientable surface of genusg ≥ 2. We use these drawings ofK n and give a polynomial-time algorithm for drawing any graph withn vertices andm edges withO(m 2 log2 g/g) many crossings on an orientable or nonorientable surface of genusg ≥ 2. Moreover, we derive lower bounds on the crossing number of any graph on a surface of genusg ≥ 0. The number of crossings in the drawings produced by our algorithm are within a multiplicative factor ofO(log2 g) from the lower bound (and hence from the optimal) for any graph withm ≥ 8n andn 2/m ≤g ≤m/64.
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References
Aggarwal, A., Klawe, M., and Shor, P., Multilayer grid embeddings for VLSI,Algorithmica,6 (1991), 129–143.
Awerbuch, B., and Leighton, T., A simple local control approximation algorithm for multicommodity flow,Proc. 34th Annual Symp. on Foundations of Computer Science, 1993, pp. 459–468.
Beineke, L. W., and Ringeisen, R. D., On the crossing number product of cycles and graphs of order four,J. Graph Theory,4 (1980), 145–155.
Brahana, H. R., Systems of circuits of two-dimensional manifolds,Ann. of Math.,30 (1923), 234–243.
Chartrand, G., and Lesniak, L.,Graphs and Digraphs, Wadsworth and Brooks/Cole, Monterey, CA, 1986.
Eades, P., and Wormald, N. C., Edge crossings in drawings of bipartite graphs,Algorithmica,11 (1994), 379–103.
Erdös, P., and Guy, R. P., Crossing number problems,Amer. Math. Monthly,80 (1973), 52–58.
Garey, M. R., and Johnson, D. S., Crossing number is NP-complete,SIAM J. Algebraic Discrete Methods,4 (1983), 312–316.
Grigoriadis, M. D., and Khachiyan, L., Fast approximation schemes for convex programs,SIAM J. Optim.,4 (1994), 86–107.
Gross, J. L., On infinite family of octahedral crossing numbers,J. Graph Theory,2 (1978), 171–178.
Guy, R. K., and Jenkins, T., The toroidal crossing number ofK m.n ,J. Combin. Theory,6 (1969), 235–250.
Guy, R. K., Jenkins, T., and Schaer, J., The toroidal crossing number of the complete graph,J. Combin. Theory,4 (1968), 376–390.
Kainen, P. C., A lower bound for crossing number of graphs with applications toK n ,K p.q , andQ(d), J. Combin. Theory Ser. B,12 (1972), 287–298.
Kainen, P. C., and White, A. T., On stable crossing numbers,J. Graph Theory,2 (1978), 181–187.
Klein, P., Plotkin, S., Stein, C., and Tardos, E., Faster approximation algorithms for unit capacity concurrent flow problems with applications to routing and sparsest cuts,SIAM J. Comput.,3(23) (1994), 466–488.
Kleitman, D. J., The crossing number ofK 5.n ,J. Combin. Theory,9 (1970), 315–323.
Leighton, F. T.,Complexity Issues in VLSI, MIT Press, Cambridge, MA, 1983.
Leighton, T., Makedon, F., Plotkin S., Stein, C., Tardos, E., and Tragoudas, S., Fast approximation algorithms for multicommodity flow problems,Proc. 23rd ACM Symp. on Theory of Computing, 1991, pp. 101–111.
Pica, G., Pisanski, T., and Ventre, A. G. S., Cartesian products of graphs and their crossing numbers,Ann. Discrete Math.,30 (1986), 339–346.
Shahrokhi, F., and Matula, D. W., The maximum concurrent flow problem,J. Assoc. Comput. Mach.,2(37) (1990), 318–334.
Shahrokhi, F., and Székely, L. A., Effective lower bounds for crossing number, bisection width and balanced vertex separators in terms of symmetry, in:Proc. Integer Programming and Combinatorial Optimization, eds. E. Balas, G. Cournèjols, R. Kannan, CMU Press, Pittsburgh, PA, 1992, pp. 102–113.
Shahrokhi, F., and Székely, L. A., On canonical concurrent flows, crossing number and graph expansion,Combin. Probab. Comput.,3 (1994), 523–543.
Shahrokhi, F., Sýkora, O., Székely, L. A., and Vrt'o, I., The crossing number of a graph on a compact 2-manifold,Adv. in Math., to appear.
Sýkora, O., and Vrt'o, I., On the crossing number of hypercubes and cube connected cycles,BIT,33 (1993), 232–237.
Szegedy, M., On plane crossing number, Unpublished manuscript, 1993.
Szegedy, M., Private communication with F. Shahrokhi, 1993.
White, A. T.,Graphs, Groups, and Surfaces, North-Holland, Amsterdam, 1984.
White, A. T., and Beineke, L. W., Topological graph theory, in:Selected Topics in Graph Theory, eds. L. W. Beineke and R. J. Wilson, Academic Press, New York, 1978, pp. 15–50.
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Communicated by G. Di Battista and R. Tamassia.
The research of the third and the fourth authors was partially supported by Grant No. 2/1138/94 of the Slovak Academy of Sciences and by EC Cooperative action IC1000 “Algorithms for Future Technologies” (Project ALTEC). A preliminary version of this paper was presented at WG93 and published in Lecture Notes in Computer Science, Vol. 790, 1993, pp. 388–396.
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Shahrokhi, F., Székely, L.A., Sýkora, O. et al. Drawings of graphs on surfaces with few crossings. Algorithmica 16, 118–131 (1996). https://doi.org/10.1007/BF02086611
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DOI: https://doi.org/10.1007/BF02086611