Simultaneous Embedding of Colored Graphs

Abstract

A set of colored graphs are compatible, if for every color i, the number of vertices of color i is the same in every graph. A simultaneous embedding of k compatibly colored graphs, each with n vertices, consists of k planar polyline drawings of these graphs such that the vertices of the same color are mapped to a common set of vertex locations. We prove that simultaneous embedding of \(k\in o(\log \log n)\) colored planar graphs, each with n vertices, can always be computed with a sublinear number of bends per edge. Specifically, we show an \(O(\min \{c, n^{1-1/\gamma }\})\) upper bound on the number of bends per edge, where \(\gamma = 2^{\lceil k/2 \rceil }\) and c is the total number of colors. Our bound, which results from a better analysis of a previously known algorithm by Durocher and Mondal (SIAM J Discrete Math 32(4):2703–2719, 2018), improves the bound for k, as well as the bend complexity by a factor of \(\sqrt{2}^{k}\). The algorithm can be generalized to obtain small universal point sets for colored graphs. We prove that \(n\lceil c/b \rceil \) vertex locations, where \(b\ge 1\), suffice to embed any set of compatibly colored n-vertex planar graphs with bend complexity O(b), where c is the number of colors.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

References

  1. 1.

    Bläsius, T., Kobourov, S.G., Rutter, I.: Simultaneous embedding of planar graphs. In: Tamassia, R. (ed.) Handbook on graph drawing and visualization, pp. 349–381. Chapman and Hall/CRC, Boca Raton (2013)

    Google Scholar 

  2. 2.

    Brandes, U., Erten, C., Estrella-Balderrama, A., Fowler, J.J., Frati, F., Geyer, M., Gutwenger, C., Hong, S., Kaufmann, M., Kobourov, S.G., Liotta, G., Mutzel, P., Symvonis, A.: Colored simultaneous geometric embeddings and universal pointsets. Algorithmica 60(3), 569–592 (2011)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Braß, P., Cenek, E., Duncan, C.A., Efrat, A., Erten, C., Ismailescu, D., Kobourov, S.G., Lubiw, A., Mitchell, J.S.B.: On simultaneous planar graph embeddings. Comput. Geom. 36(2), 117–130 (2007)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Di Giacomo, E., Didimo, W., Liotta, G., Meijer, H., Trotta, F., Wismath, S.K.: \(k\)-colored point-set embeddability of outerplanar graphs. J. Graph Algorithms Appl. 12(1), 29–49 (2008)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Di Giacomo, E., Didimo, W., Liotta, G., Wismath, S.K.: Curve-constrained drawings of planar graphs. Comput. Geom. 30(1), 1–23 (2005)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Di Giacomo, E., Gasieniec, L., Liotta, G., Navarra, A.: Colored point-set embeddings of acyclic graphs. In: Frati, F., Ma, K. (eds.) Proceedings of the 25th international symposium on graph drawing and network visualization, LNCS, vol. 10692, pp. 413–425. Springer (2018)

  7. 7.

    Di Giacomo, E., Liotta, G.: Simultaneous embedding of outerplanar graphs, paths, and cycles. Int. J. Comput. Geom. Appl. 17(2), 139–160 (2007)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Durocher, S., Mondal, D.: Relating graph thickness to planar layers and bend complexity. SIAM J. Discrete Math. 32(4), 2703–2719 (2018)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Erdős, P., Szekeres, G.: A combinatorial problem in geometry. Compos. Math. 2, 463–470 (1935)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Erten, C., Kobourov, S.G.: Simultaneous embedding of planar graphs with few bends. J. Graph Algorithms Appl. 9(3), 347–364 (2005)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Everett, H., Lazard, S., Liotta, G., Wismath, S.K.: Universal sets of \(n\) points for one-bend drawings of planar graphs with \(n\) vertices. Discret. Comput. Geom. 43(2), 272–288 (2010)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Fáry, I.: On straight-line representation of planar graphs. Acta Sci. Math. 11, 229–233 (1948)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Mondal, D., Mondal, M., Roy, C.K., Schneider, K.A., Li, Y., Wang, S.: Clone-world: a visual analytic system for large scale software clones. Vis. Inform. 3(1), 18–26 (2019)

    Article  Google Scholar 

  14. 14.

    Pach, J., Wenger, R.: Embedding planar graphs at fixed vertex locations. Graphs Comb. 17(4), 717–728 (2001)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Tahmasbi, M., Hashemi, S.M.: Orthogonal thickness of graphs. In: Proceedings of the 22nd annual Canadian conference on computational geometry (CCCG), pp. 199–202. (2010)

Download references

Acknowledgements

The author thanks the anonymous reviewers for their helpful comments, which improved the presentation of the paper.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Debajyoti Mondal.

Additional information

Publisher's Note

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

This work is supported in part by Natural Sciences and Engineering Research Council of Canada (NSERC).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Mondal, D. Simultaneous Embedding of Colored Graphs. Graphs and Combinatorics (2021). https://doi.org/10.1007/s00373-021-02276-y

Download citation

Keywords

  • Graph drawing
  • Simultaneous embedding
  • Monotonic sequence
  • Bend complexity
  • Thickness