Advertisement

Crossing Number for Graphs with Bounded Pathwidth

  • 7 Accesses

Abstract

The crossing number is the smallest number of pairwise edge crossings when drawing a graph into the plane. There are only very few graph classes for which the exact crossing number is known or for which there at least exist constant approximation ratios. Furthermore, up to now, general crossing number computations have never been successfully tackled using bounded width of graph decompositions, like treewidth or pathwidth. In this paper, we show that the crossing number is tractable (even in linear time) for maximal graphs of bounded pathwidth 3. The technique also shows that the crossing number and the rectilinear (a.k.a. straight-line) crossing number are identical for this graph class, and that we require only an \(O(n)\times O(n)\)-grid to achieve such a drawing. Our techniques can further be extended to devise a 2-approximation for general graphs with pathwidth 3. One crucial ingredient here is that the crossing number of a graph with a separation pair can be lower-bounded using the crossing numbers of its cut-components, a result that may be interesting in its own right. Finally, we give a \(4{\mathbf{w}}^3\)-approximation of the crossing number for maximal graphs of pathwidth \({\mathbf{w}}\). This is a constant approximation for bounded pathwidth. We complement this with an NP-hardness proof of the weighted crossing number already for pathwidth 3 graphs and bicliques \(K_{3,n}\).

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

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

Subscribe to journal

Immediate online access to all issues from 2019. Subscription will auto renew annually.

US$ 199

This is the net price. Taxes to be calculated in checkout.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Notes

  1. 1.

    The coordinates are chosen to be easy to define and analyze; the constant factor could likely be improved by making more careful choices.

References

  1. 1.

    Ajtai, M., Chvátal, V., Newborn, M.M., Szemerédi, E.: Crossing-free subgraphs. In: Hammer, P.L., Rosa, A., Sabidussi, G., Turgeon, J. (ed.) Theory and Practice of Combinatorics, Volume 60 of North-Holland Mathematics Studies, pp 9 – 12. North-Holland (1982). https://www.sciencedirect.com/science/article/pii/S0304020808734844

  2. 2.

    Biedl, T., Chimani, M., Derka, M., Mutzel, P.: Crossing number for graphs with bounded pathwidth. In: ISAAC ’07, LIPIcs, pp. 13:1–13:13 (2017). https://doi.org/10.4230/LIPIcs.ISAAC.2017.13

  3. 3.

    Bienstock, D.: Some provably hard crossing number problems. Discrete Comput. Geom. 6, 443–459 (1991)

  4. 4.

    Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25(6), 1305–1317 (1996)

  5. 5.

    Bodlaender, H.L., Kloks, T.: Efficient and constructive algorithms for the pathwidth and treewidth of graphs. J. Algorithms 21(2), 358–402 (1996)

  6. 6.

    Bokal, D.: On the crossing numbers of cartesian products with paths. J. Comb. Theory Ser. B 97(3), 381–384 (2007)

  7. 7.

    Cabello, S.: Hardness of approximation for crossing number. Discrete Comput. Geom. 49(2), 348–358 (2013)

  8. 8.

    Cabello, S., Mohar, B.: Crossing number and weighted crossing number of near-planar graphs. Algorithmica 60(3), 484–504 (2011)

  9. 9.

    Chimani, M., Hliněný, P.: A tighter insertion-based approximation of the crossing number. J. Comb. Optim. 33, 1–43 (2016)

  10. 10.

    Chimani, M., Hliněný, P.: Inserting multiple edges into a planar graph. In: SoCG 2016, LIPIcs, pp. 30:1–30:15 (2016). https://doi.org/10.4230/LIPIcs.SoCG.2016.30

  11. 11.

    Chimani, M., Hliněný, P., Mutzel, P.: Vertex insertion approximates the crossing number for apex graphs. Eur. J. Comb. 33, 326–335 (2012)

  12. 12.

    Chuzhoy, J.: An algorithm for the graph crossing number problem. In: STOC ’11, pp. 303–312. ACM (2011)

  13. 13.

    Courcelle, B.: The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inf. Comput. 85(1), 12–75 (1990)

  14. 14.

    de Klerk, E., Maharry, J., Pasechnik, D.V., Richter, R.B., Salazar, G.: Improved bounds for the crossing numbers of \(K_{m, n}\) and \(K_n\). SIAM J. Discrete Math. 20(1), 189–202 (2006)

  15. 15.

    de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10, 41–51 (1990)

  16. 16.

    Fox, J., Pach, J., Suk, A.: Approximating the rectilinear crossing number. In: GD 2016, LNCS 9801, pp. 413–426. Springer (2016)

  17. 17.

    Gitler, I., Hliněný, P., Leanos, J., Salazar, G.: The crossing number of a projective graph is quadratic in the face-width. Electron. J. Comb 15(1), #R46 (2008)

  18. 18.

    Grohe, M.: Computing crossing numbers in quadratic time. J. Comput. Syst. Sci. 68(2), 285–302 (2004)

  19. 19.

    Hliněný, P.: Crossing-number critical graphs have bounded path-width. J. Comb. Theory Ser. B 88(2), 347–367 (2003)

  20. 20.

    Hliněný, P., Chimani, M.: Approximating the crossing number of graphs embeddable in any orientable surface. In: SODA ’10, pp. 918–927 (2010)

  21. 21.

    Hliněný, P., Salazar, G.: Approximating the crossing number of toroidal graphs. In: ISAAC ’07, LNCS 4835, pp. 148–159. Springer (2007)

  22. 22.

    Kawarabayashi, K.-I., Reed, B.: Computing crossing number in linear time. In: STOC ’07, pp. 382–390 (2007)

  23. 23.

    Kleitman, D.J.: The crossing number of \({K}_{5, n}\). J. Comb. Theory 9(4), 315–323 (1970)

  24. 24.

    Klešč, M., Petrillová, J.: The crossing numbers of products of path with graphs of order six. Discuss. Math. Graph Theory 33(3), 571–582 (2013)

  25. 25.

    Kloks, T.: Treewidth, Computations and Approximations. LNCS 842. Springer, Berlin (1994)

  26. 26.

    Pan, S., Richter, R.B.: The crossing number of \({K}_{11}\) is 100. J. Graph Theory 56(2), 128–134 (2007)

  27. 27.

    Richter, R.B., Salazar, G.: The crossing number of \({P}({N},3)\). Graphs Comb. 18(2), 381–394 (2002)

  28. 28.

    Schaefer, M.: Crossing Numbers of Graphs. CRC Press, Boca Raton (2017)

  29. 29.

    Schnyder, W.: Embedding planar graphs on the grid. In: ACM-SIAM Symposium on Discrete Algorithms (SODA ’90), pp. 138–148 (1990)

  30. 30.

    Singer, D.A.: The rectilinear crossing number of certain graphs (1971). http://www.cwru.edu/artsci/math/singer/publish/Rectilinear_crossings.pdf

  31. 31.

    Tamassia, R.: Handbook of Graph Drawing and Visualization. CRC Press, Boca Raton (2013)

  32. 32.

    Vrt’o, I.: Crossing numbers of graphs: a bibliography (2014). ftp://ftp.ifi.savba.sk/pub/imrich/crobib.pdf

  33. 33.

    Wood, D.R., Telle, J.A.: Planar decompositions and the crossing number of graphs with an excluded minor. N. Y. J. Math. 13, 117–146 (2007)

Download references

Author information

Correspondence to Markus Chimani.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work has been started at the Crossing Number Workshop 2016 in Strobl (Austria) and appeared in a preliminary version at ISAAC’17 [2]. Research of T.B. supported by NSERC. Research of M.D. supported by NSERC Vanier CSG. Research of M.C. partially supported by the German Science Foundation (DFG), project CH 897/2-1. Research of P.M. partially supported by the DFG within the SFB 876 (project A6).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Biedl, T., Chimani, M., Derka, M. et al. Crossing Number for Graphs with Bounded Pathwidth. Algorithmica (2020). https://doi.org/10.1007/s00453-019-00653-x

Download citation

Keywords

  • Crossing number
  • Pathwidth
  • Approximation
  • Graph algorithms
  • Complexity