Bounded Embeddings of Graphs in the Plane

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9843)


A drawing in the plane (\(\mathbb {R}^2\)) of a graph \(G=(V,E)\) equipped with a function \(\gamma : V \rightarrow \mathbb {N}\) is x -bounded if (i) \(x(u) <x(v)\) whenever \(\gamma (u)<\gamma (v)\) and (ii) \(\gamma (u)\le \gamma (w)\le \gamma (v)\), where \(uv\in E\) and \(\gamma (u)\le \gamma (v)\), whenever \(x(w)\in x(uv)\), where x(.) denotes the projection to the x-axis. We prove a characterization of isotopy classes of embeddings of connected graphs equipped with \(\gamma \) in the plane containing an x-bounded embedding. Then we present an efficient algorithm, which relies on our result, for testing the existence of an x-bounded embedding if the given graph is a forest. This partially answers a question raised recently by Angelini et al. and Chang et al., and proves that c-planarity testing of flat clustered graphs with three clusters is tractable when the underlying abstract graph is a forest.


Graph planarity testing Weakly simple embedding c-planarity PQ-tree Algebraic crossing number 


  1. 1.
    Angelini, P., Da Lozzo, G., Di Battista, G., Frati, F.: Strip planarity testing. In: Wismath, S., Wolff, A. (eds.) GD 2013. LNCS, vol. 8242, pp. 37–48. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  2. 2.
    Angelini, P., Da Lozzo, G., Di Battista, G., Di Donato, V., Kindermann, P., Rote, G., Rutter, I., Planarity, W.: Embedding graphs with direction-constrained edges, Chap. 70, pp. 985–996 (2016)Google Scholar
  3. 3.
    Bachmaier, C., Brandenburg, F.J., Forster, M.: Radial level planarity testing and embedding in linear time. In: Liotta, G. (ed.) GD 2003. LNCS, vol. 2912, pp. 393–405. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  4. 4.
    Bertolazzi, P., Di Battista, G., Liotta, G., Mannino, C.: Upward drawings of triconnected digraphs. Algorithmica 12(6), 476–497 (1994)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Biedl, T.C.: Drawing planar partitions III: two constrained embedding problems. Rutcor Res. Rep. 13–98, 13–98 (1998)Google Scholar
  6. 6.
    Bläsius, T., Rutter, I.: Simultaneous PQ-ordering with applications to constrained embedding problems. In: Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013, New Orleans, 6–8 January 2013, pp. 1030–1043 (2013).
  7. 7.
    Booth, K.S., Lueker, G.S.: Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms. J. Comput. Syst. Sci. 13(3), 335–379 (1976)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Brass, P., Cenek, E., Duncan, C.A., Efrat, A., Erten, C., Ismailescu, D.P., Kobourov, S.G., Lubiw, A., Mitchell, J.S.B.: On simultaneous planar graph embeddings. Comput. Geometry 36(2), 117–130 (2007)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Cairns, G., Nikolayevsky, Y.: Bounds for generalized thrackles. Discrete Comput. Geom. 23(2), 191–206 (2000)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Chang, H.C., Erickson, J., Xu, C.: Detecting weakly simple polygons. In: Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1655–1670 (2015). arXiv:1407.3340
  11. 11.
    Cortese, P.F., Di Battista, G., Frati, F., Patrignani, M., Pizzonia, M.: C-planarity of c-connected clustered graphs. J. Graph Algorithms Appl. 12(2), 225–262 (2008)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Cortese, P.F., Di Battista, G., Patrignani, M., Pizzonia, M.: Clustering cycles into cycles of clusters. J. Graph Algorithms Appl. 9(3), 391–413 (2005)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Cortese, P.F., Di Battista, G., Patrignani, M., Pizzonia, M.: On embedding a cycle in a plane graph. Discrete Math. 309(7), 1856–1869 (2009)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Di Battista, G., Tamassia, R.: Incremental planarity testing. In: 30th Annual Symposium on Foundations of Computer Science, pp. 436–441, October 1989Google Scholar
  15. 15.
    Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173, 3rd edn. Springer, Heidelberg (2005)MATHGoogle Scholar
  16. 16.
    Feng, Q.-W., Cohen, R.F., Eades, P.: How to draw a planar clustered graph. In: Ding-Zhu, D., Li, M. (eds.) COCOON 1995. LNCS, vol. 959, pp. 21–30. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  17. 17.
    Feng, Q.-W., Cohen, R.F., Eades, P.: Planarity for clustered graphs. In: Spirakis, P. (ed.) Algorithms–ESA 1995. LNCS, vol. 979, pp. 213–226. Springer, Heidelberg (1995)Google Scholar
  18. 18.
    Radoslav Fulek. Toward the Hanani-Tutte theorem for clustered graphs. 2014. arXiv:1410.3022v2
  19. 19.
    Fulek, R.: Towards the Hanani-Tutte theorem for clustered graphs. In: Kratsch, D., Todinca, I. (eds.) WG 2014. LNCS, vol. 8747, pp. 176–188. Springer, Heidelberg (2014)Google Scholar
  20. 20.
    Fulek, R.: C-planarity of embedded cyclic c-graphs (2016). arXiv:1602.01346v2
  21. 21.
    Fulek, R., Pelsmajer, M.J., Schaefer, M., Štefankovič, D.: Hanani-Tutte, monotone drawings and level-planarity. In: Pach, J. (ed.) Thirty Essays on Geometric Graph Theory, pp. 263–288. Springer, New York (2012)Google Scholar
  22. 22.
    Garg, A., Tamassia, R.: On the computational complexity of upward and rectilinear planarity testing. SIAM J. Comput. 31(2), 601–625 (2002)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Goodrich, M.T., Lueker, G.S., Sun, J.Z.: C-planarity of extrovert clustered graphs. In: Healy, P., Nikolov, N.S. (eds.) GD 2005. LNCS, vol. 3843, pp. 211–222. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  24. 24.
    Gutwenger, C., Jünger, M., Leipert, S., Mutzel, P., Percan, M., Weiskircher, R.: Advances in c-planarity testing of clustered graphs. In: Goodrich, M.T., Kobourov, S.G. (eds.) GD 2002. LNCS, vol. 2528. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  25. 25.
    Hong, S.-H., Nagamochi, H.: Simpler algorithms for testing two-page book embedding of partitioned graphs. Theoretical Computer Science (2016)Google Scholar
  26. 26.
    Hsu, W.-L., McConnell, R.M.: PC-trees and circular-ones arrangements. Theoret. Comput. Sci. 296(1), 99–116 (2003)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Jelínek, V., Jelínková, E., Kratochvíl, J., Lidický, B.: Clustered planarity: embedded clustered graphs with two-component clusters. In: Tollis, I.G., Patrignani, M. (eds.) GD 2008. LNCS, vol. 5417, pp. 121–132. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  28. 28.
    Jelınková, E., Kára, J., Kratochvıl, J., Pergel, M., Suchỳ, O., Vyskocil, T.: Clustered planarity: small clusters in cycles and Eulerian graphs. J. Graph Algorithms Appl. 13(3), 379–422 (2009)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Jünger, Michael, Leipert, Sebastian, Mutzel, Petra: Level planarity testing in linear time. In: Whitesides, Sue H. (ed.) GD 1998. LNCS, vol. 1547, pp. 224–237. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  30. 30.
    Lengauer, T.: Hierarchical planarity testing algorithms. J. ACM 36(3), 474–509 (1989)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Mabillard, I., Wagner, U. Eliminating Tverberg points, I. An analogue of the Whitney trick. In: Proceedings of theThirtieth Annual Symposium on Computational Geometry, SOCG 2014, pp. 171:171–171:180 (2014)Google Scholar
  32. 32.
    Opatrny, J.: Total ordering problem. SIAM J. Comput. 8(1), 111–114 (1979)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Pach, J., Tóth, G.: Monotone drawings of planar graphs. J. Graph Theory 46(1), 39–47 (2004). arXiv:1101.0967 MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Schaefer, M.: Toward a theory of planarity: Hanani-Tutte and planarity variants. J. Graph Algorithms Appl. 17(4), 367–440 (2013)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Ziegler, G.M.: Lectures on Polytopes, vol. 152. Springer Science & Business, New York (1995)MATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.IST AustriaKlosterneuburgAustria

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