International Symposium on Graph Drawing and Network Visualization

Graph Drawing and Network Visualization pp 125-138 | Cite as

The Book Embedding Problem from a SAT-Solving Perspective

  • Michael A. Bekos
  • Michael Kaufmann
  • Christian Zielke
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9411)

Abstract

In a book embedding, the vertices of a graph are placed on the spine of a book and the edges are assigned to pages, so that edges of the same page do not cross. In this paper, we approach the problem of determining whether a graph can be embedded in a book of a certain number of pages from a different perspective: We propose a simple and quite intuitive SAT formulation, which is robust enough to solve non-trivial instances of the problem in reasonable time. As a byproduct, we show a lower bound of 4 on the page number of 1-planar graphs.

References

  1. 1.
    Bekos, M., Gronemann, M., Raftopoulou, C.N.: Two-page book embeddings of 4-planar graphs. In: STACS, vol. 25, pp. 137–148. LIPIcs, Schloss Dagstuhl (2014)Google Scholar
  2. 2.
    Bekos, M.A., Bruckdorfer, T., Kaufmann, M., Raftopoulou, C.: 1-planar graphs have constant book thickness. In: Bansal, N., Finocchi, I. (eds.) ESA 2015. LNCS, vol. 9294, pp. 130–141. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  3. 3.
    Bernhart, F., Kainen, P.: The book thickness of a graph. Comb. Theory 27(3), 320–331 (1979)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Biedl, T., Bläsius, T., Niedermann, B., Nöllenburg, M., Prutkin, R., Rutter, I.: Using ILP/SAT to determine pathwidth, visibility representations, and other grid-based graph drawings. In: Wismath, S., Wolff, A. (eds.) GD 2013. LNCS, vol. 8242, pp. 460–471. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  5. 5.
    Bilski, T.: Embedding graphs in books: a survey. IEEE Proc. Comput. Digit. Tech. 139(2), 134–138 (1992)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Blankenship, R.: Book embeddings of graphs. Ph.D. thesis, Louisiana State University (2003)Google Scholar
  7. 7.
    Bodendiek, R., Schumacher, H., Wagner, K.: Über 1-optimale graphen. Math. Nachr. 117(1), 323–339 (1984)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Brinkmann, G., Greenberg, S., Greenhill, C.S., McKay, B.D., Thomas, R., Wollan, P.: Generation of simple quadrangulations of the sphere. Discrete Math. 305(1–3), 33–54 (2005)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Cheeseman, P., Kanefsky, B., Taylor, W.M.: Where the really hard problems are. In: Mylopoulos, J., Reiter, R. (eds.) AI, pp. 331–340. Morgan Kaufmann (1991)Google Scholar
  10. 10.
    Chimani, M., Zeranski, R.: Upward planarity testing via SAT. In: Didimo, W., Patrignani, M. (eds.) GD 2012. LNCS, vol. 7704, pp. 248–259. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  11. 11.
    Chung, F.R.K., Leighton, F.T., Rosenberg, A.L.: Embedding graphs in books: a layout problem with applications to VLSI design. SIAM J. Algebraic Discrete Method 8(1), 33–58 (1987)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Cornuéjols, G., Naddef, D., Pulleyblank, W.: Halin graphs and the travelling salesman problem. Math. Programm. 26(3), 287–294 (1983)CrossRefMATHGoogle Scholar
  13. 13.
    Dujmović, V., Wood, D.: Graph treewidth and geometric thickness parameters. Discrete Comput. Geom. 37(4), 641–670 (2007)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Eén, N., Sörensson, N.: An extensible SAT-solver. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 502–518. Springer, Heidelberg (2004) CrossRefGoogle Scholar
  15. 15.
    Gange, G., Stuckey, P.J., Marriott, K.: Optimal k-level planarization and crossing minimization. In: Brandes, U., Cornelsen, S. (eds.) GD 2010. LNCS, vol. 6502, pp. 238–249. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  16. 16.
    Ganley, J.L., Heath, L.S.: The pagenumber of \(k\)-trees is \(O(k)\). Discrete Appl. Math. 109(3), 215–221 (2001)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Goldner, A., Harary, F.: Note on a smallest nonhamiltonian maximal planar graph. Bull. Malays. Math. Sci. Soc. 1(6), 41–42 (1975)Google Scholar
  18. 18.
    Heath, L.: Embedding planar graphs in seven pages. In: FOCS, pp. 74–83. IEEE Computer Society (1984)Google Scholar
  19. 19.
    Heath, L.: Algorithms for embedding graphs in books. Ph.D. thesis, University of N. Carolina (1985)Google Scholar
  20. 20.
    Heath, L.S., Leighton, F.T., Rosenberg, A.L.: Comparing queues and stacks as machines for laying out graphs. SIAM J. Discrete Math. 3(5), 398–412 (1992)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Kainen, P.C., Overbay, S.: Extension of a theorem of Whitney. Appl. Math. Lett. 20(7), 835–837 (2007)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Kottler, S.: Description of the SApperloT, SArTagnan and MoUsSaka solvers for the SAT-competition 2011 (2011)Google Scholar
  23. 23.
    Malitz, S.: Genus \(g\) graphs have pagenumber \(O(\sqrt{q})\). J. Algorithms 17(1), 85–109 (1994)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Malitz, S.: Graphs with \(e\) edges have pagenumber \(O(\sqrt{E})\). J. Algorithms 17(1), 71–84 (1994)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Nishizeki, T., Chiba, N.: Hamiltonian cycles. In: Planar Graphs: Theory and Algorithms, chap. 10, pp. 171–184. Dover Books on Mathematics, Courier Dover Publications (2008)Google Scholar
  26. 26.
    Ollmann, T.: On the book thicknesses of various graphs. In: Hoffman, F., Levow, R., Thomas, R. (eds.) Southeastern Conference on Combinatorics, Graph Theory and Computing. Congressus Numerantium, vol. VIII, p. 459 (1973)Google Scholar
  27. 27.
    Plaisted, D.A., Greenbaum, S.: A structure-preserving clause form translation. J. Symbolic Comput. 2(3), 293–304 (1986)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Rosenberg, A.L.: The Diogenes approach to testable fault-tolerant arrays of processors. IEEE Trans. Comput. C–32(10), 902–910 (1983)CrossRefGoogle Scholar
  29. 29.
    Suzuki, Y.: Optimal 1-planar graphs which triangulate other surfaces. Discrete Math. 310(1), 6–11 (2010)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Tarjan, R.: Sorting using networks of queues and stacks. J. ACM 19(2), 341–346 (1972)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Velev, M.N., Gao, P.: Efficient SAT techniques for relative encoding of permutations with constraints. In: Nicholson, A., Li, X. (eds.) AI 2009. LNCS, vol. 5866, pp. 517–527. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  32. 32.
    Wigderson, A.: The complexity of the Hamiltonian circuit problem for maximal planar graphs. Technical report TR-298, EECS Department, Princeton University (1982)Google Scholar
  33. 33.
    Yannakakis, M.: Embedding planar graphs in four pages. J. Comput. Syst. Sci. C–38(1), 36–67 (1989)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Michael A. Bekos
    • 1
  • Michael Kaufmann
    • 1
  • Christian Zielke
    • 1
  1. 1.Wilhelm-Schickard-Institut Für InformatikUniversität TübingenTübingenGermany

Personalised recommendations