Skip to main content
SpringerLink
Log in

Book embeddings and crossing numbers

  • Conference paper
  • First Online:
Graph-Theoretic Concepts in Computer Science (WG 1994)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 903))

Included in the following conference series:

Abstract

The paper introduces the book crossing number problem which can be viewed as a variant of the well-known plane and surface crossing number problem or as a generalization of the book embedding problem. The book crossing number of a graph G is defined as the minimum number of edge crossings when the vertices of G are placed on the spine of a k-page book and edges are drawn on pages, so that each edge is contained by one page. We present polynomial time algorithms for drawing graphs in books with small number of crossings. One algorithm is suitable for sparse graphs and gives a drawing in which the number of crossings is within a multiplicative factor of O(log2 n) from the optimal one under certain conditions. Using these drawings we improve the best known upper bound on the rectilinear crossing number, provided that m≥4n. We also derive a general lower bound on the book crossing number of any graph and present a second polynomial time algorithm to generate a drawing of any graph with O(m 2/k 2) many edge crossings. This number of crossings is within a constant multiplicative factor from our general lower bound of Ω(m 3/n 2 k 2), provided that m=Θ(n 2). For several classes of well-known graphs, we also sharpen our algorithmic upper bounds by giving specific drawings.

This research was partially supported by EC Cooperative Action IC1000 “Project ALTEC” and the Slovak Academy of Sciences Grant No. 2/1138/94

This research was supported by the European Community programme “COST” and done at Laboratory for Computer Science, University of Paris XI

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ajtai, M., Chvátal, V., Newborn, M.M., Szemerédy, E., “Crossing-free subgraphs”, Annals of Discrete Mathematics 12 (1982), 9–12.

    Google Scholar 

  2. Alon, N., Spencer, J.H., Erdős, P., “The Probabilistic Method”, Wiley and Sons, New York, 1992.

    Google Scholar 

  3. Behzad, M., Chartrand, G., Lesniak-Foster, L., “Graphs and Digraphs”, Wadsworth International Group, Belmont, 1976.

    Google Scholar 

  4. Bernhart, F., Kainen, P. C., “The book thickness of a graph”, J. Combinatorial Theory, Series B 27 (1979), 320–331.

    Google Scholar 

  5. Bienstock, D., Dean, N., “New results on rectilinear crossing Numbers and Plane Embeddings”, J. Graph Theorey, 16 (1992), 389–398.

    Google Scholar 

  6. Chinn, P. Z., Chvátalová, L., Dewdney, A. K., Gibbs, N. E., “The bandwidth problem for graphs and matrices—a survey”, J. Graph Theory 6 (1982), 223–253.

    Google Scholar 

  7. Chung, F. R. K., “Labeling of graphs”, in: “Selected Topics in Graph Theory 3”, (L. Beineke and R. Wilson, eds.), Academic Press, New York, 1988, 151–168.

    Google Scholar 

  8. Chung, F. R. K., Leighton, F. T., Rosenberg, A. L., “Embeddings graphs in books: A layout problem with applications to VLSI design”, SIAM J. Algebraic and Discrete Methods 8 (1987), 33–58.

    Google Scholar 

  9. Chung, F. R. K., Yau, S.T., “A near optimal algorithm for edge separators”, in: Proc. 28th ACM Annual Symposium on Theory of Computing, ACM Press, 1994, 1–8.

    Google Scholar 

  10. Díaz, J., “Graph layout problems”, in: Proc. 17th Intl. Symposium on Mathematical Foundations of Computer Science, LNCS 629, Springer Verlag, Berlin, 1992, 15–23.

    Google Scholar 

  11. Erdős, P., Guy, R. P., “Crossing number problems”, American Mathematical Monthly 80 (1973), 52–58.

    Google Scholar 

  12. Guy, R. P., Jenkyns, T., Schaer, J., “The toroidal crossing number of the complete graph”, J. Combinatorial Theory 4 (1968), 376–390.

    Google Scholar 

  13. Kainen, P. C., “The book thickness of a graph, II”, Congressus Numerantium 71 (1990), 127–132.

    Google Scholar 

  14. Leighton, F. T., “Complexity Issues in VLSI”, M.I.T. Press, Cambridge, 1983.

    Google Scholar 

  15. Leiserson, C. E., “Area efficient graph layouts (for VLSI)”, in: Proc. 21st Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, 1980, 270–281.

    Google Scholar 

  16. Masuda, S., Kashiwabara, T., Nakajima, K., Fujisawa, T., “On the NP-completeness of a computer network layout problem”, in: Proc. 1987 IEEE Intl. Symposium on Circuits and Systems, IEEE Computer Society Press, Los Alamitos, 1987, 292–295.

    Google Scholar 

  17. Masuda, S., Kashiwabara, T., Nakajima, K., Fujisawa, T., “Crossing minimization in linear embeddings of graphs”, IEEE Transactions on Computers 39 (1990), 124–127.

    Google Scholar 

  18. Melikhov, A. N., Koreichik, V. M., Tishchenko, V. A., “Minimization of the number of intersections of edges of a graph”, (in Russian), Vichislityelnie Sistemi Vip. 41 (1971), 32–40.

    Google Scholar 

  19. Muder, J. D., Weawer, M. L., West, D. B., “Pagenumber of complete bipartite graphs”, J. Graph Theory 12 (1988), 469–489.

    Google Scholar 

  20. Nicolson, T. A. J., “Permutation procedure for minimizing the number of crossings in a network”, Proc. Inst. Elec. Engnrs. 115 (1968), 21–26.

    Google Scholar 

  21. Pach, J., Shahrokhi, F., Szegedy, M., “Applications of crossing numbers”, in: Proc. 10th Annual ACM Symposium on Computational Geometry, ACM Press, New York, 1994.

    Google Scholar 

  22. Shahrokhi, F., 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, CMU Press, Pittsburgh, 1992, 102–113.

    Google Scholar 

  23. Shahrokhi, F., Sýkora, O., Székely, L. A., Vrt'o, I., “The crossing number of a graph on a compact 2-manifold”, Advances in Mathematics, to appear.

    Google Scholar 

  24. Shahrokhi, F., Sykora, O., Székely, L. A., Vrt'o, I., “Improved bounds for the crossing numbers on surfaces of genus g”, in: Proc. 19-th Intl. Workshop on Graph-Theoretic Concepts in Computer Science WG'93, LNCS 790, Springer Verlag, Berlin, 1994, 388–397.

    Google Scholar 

  25. Sýkora, O., Vrt'o, I., “On crossing numbers of hypercubes and cube connected cycles”, BIT 3 (1993), 232–237.

    Google Scholar 

  26. Ullman, J. D., “Computational Aspects of VLSI”, Computer Science Press, Rockville, 1984.

    Google Scholar 

  27. Yannakakis, M., “Linear and book embeddings of graphs”, in: Proc. Aegean Workshop on Computing, LNCS 227, Springer Verlag, Berlin, 1986, 229–240.

    Google Scholar 

  28. Yannakakis, M., “Four pages are necessary and sufficient for planar graphs”, in: Proc. 18th ACM Annual Symposium on Theory of Computing, ACM Press, New York, 1986, 104–108.

    Google Scholar 

  29. Zarankiewicz, K., “On a problem of P. Turán concerning graphs”, Fundamenta Mathematica 41 (1954), 137–145.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of North Texas, P.O.Box 13886, Denton, TX, USA

    Farhad Shahrokhi

  2. Institute for Informatics, Slovak Academy of Sciences, Dúbravská 9, 842 35, Bratislava, Slovak Republic

    Ondrej Sýkora & Imrich Vrt'o

  3. Department of Computer Science, Eőtvős University, Múzeum krt. 6-8, 1088, Budapest, Hungary

    László A. Székely

Authors
  1. Farhad Shahrokhi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ondrej Sýkora
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. László A. Székely
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Imrich Vrt'o
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Ernst W. Mayr Gunther Schmidt Gottfried Tinhofer

Rights and permissions

Reprints and permissions

Copyright information

© 1995 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Shahrokhi, F., Sýkora, O., Székely, L.A., Vrt'o, I. (1995). Book embeddings and crossing numbers. In: Mayr, E.W., Schmidt, G., Tinhofer, G. (eds) Graph-Theoretic Concepts in Computer Science. WG 1994. Lecture Notes in Computer Science, vol 903. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-59071-4_53

Download citation

  • DOI: https://doi.org/10.1007/3-540-59071-4_53

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-59071-2

  • Online ISBN: 978-3-540-49183-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation