Skip to main content
Springer Nature Link
Log in

Inserting One Edge into a Simple Drawing Is Hard

  • Conference paper
  • First Online:
Graph-Theoretic Concepts in Computer Science (WG 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12301))

Included in the following conference series:

  • 552 Accesses

Abstract

A simple drawing D(G) of a graph G is one where each pair of edges share at most one point: either a common endpoint or a proper crossing. An edge e in the complement of G can be inserted into D(G) if there exists a simple drawing of \(G+e\) extending D(G). As a result of Levi’s Enlargement Lemma, if a drawing is rectilinear (pseudolinear), that is, the edges can be extended into an arrangement of lines (pseudolines), then any edge in the complement of G can be inserted. In contrast, we show that it is \(\mathsf {NP}\)-complete to decide whether one edge can be inserted into a simple drawing. This remains true even if we assume that the drawing is pseudocircular, that is, the edges can be extended to an arrangement of pseudocircles. On the positive side, we show that, given an arrangement of pseudocircles \(\mathcal {A}\) and a pseudosegment \(\sigma \), it can be decided in polynomial time whether there exists a pseudocircle \(\varPhi _\sigma \) extending \(\sigma \) for which \(\mathcal {A}\cup \{\varPhi _\sigma \}\) is again an arrangement of pseudocircles.

A. Arroyo—Funded by the Marie Skłodowska-Curie grant agreement No 754411.

F. Klute—This work was conducted while Fabian Klute was a member of the “Algorithms and Complexity Group” at TU Wien.

I. Parada—Partially supported by the Austrian Science Fund (FWF): W1230 and within the collaborative DACH project Arrangements and Drawings as FWF project I 3340-N35.

B. Vogtenhuber—Partially supported by Austrian Science Fund (FWF) within the collaborative DACH project Arrangements and Drawings as FWF project I 3340-N35.

T. Wiedera—Supported by the German Research Foundation (DFG) grant CH 897/2-2.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    Also known as Levi’s Extension Lemma. Several different proofs of Levi’s Enlargement Lemma have been published since then  [3, 14, 31,32,33].

References

  1. Angelini, P., et al.: Testing planarity of partially embedded graphs. ACM Trans. Algorithms 11(4), 32:1–32:42 (2015). https://doi.org/10.1145/2629341

    Article  MathSciNet  MATH  Google Scholar 

  2. Arroyo, A., Derka, M., Parada, I.: Extending simple drawings. In: Archambault, D., Tóth, C.D. (eds.) GD 2019. LNCS, vol. 11904, pp. 230–243. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-35802-0_18

    Chapter  Google Scholar 

  3. Arroyo, A., McQuillan, D., Richter, R.B., Salazar, G.: Levi’s lemma, pseudolinear drawings of \({K}_n\), and empty triangles. J. Graph Theory 87(4), 443–459 (2018). https://doi.org/10.1002/jgt.22167

    Article  MathSciNet  MATH  Google Scholar 

  4. Bagheri, A., Razzazi, M.: Planar straight-line point-set embedding of trees with partial embeddings. Inf. Process. Lett. 110(12–13), 521–523 (2010). https://doi.org/10.1016/j.ipl.2010.04.019

    Article  MathSciNet  MATH  Google Scholar 

  5. Brückner, G., Rutter, I.: Partial and constrained level planarity. In: Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2017), pp. 2000–2011. SIAM (2017). https://doi.org/10.1137/1.9781611974782.130

  6. Cabello, S., Mohar, B.: Adding one edge to planar graphs makes crossing number and 1-planarity hard. SIAM J. Comput. 42(5), 1803–1829 (2013). https://doi.org/10.1137/120872310

    Article  MathSciNet  MATH  Google Scholar 

  7. Chaplick, S., Dorbec, P., Kratochvíl, J., Montassier, M., Stacho, J.: Contact representations of planar graphs: extending a partial representation is hard. In: Kratsch, D., Todinca, I. (eds.) WG 2014. LNCS, vol. 8747, pp. 139–151. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-12340-0_12

    Chapter  Google Scholar 

  8. Chaplick, S., Fulek, R., Klavík, P.: Extending partial representations of circle graphs. J. Graph Theory 91(4), 365–394 (2019). https://doi.org/10.1002/jgt.22436

    Article  MathSciNet  MATH  Google Scholar 

  9. Chaplick, S., Guśpiel, G., Gutowski, G., Krawczyk, T., Liotta, G.: The partial visibility representation extension problem. Algorithmica 80(8), 2286–2323 (2017). https://doi.org/10.1007/s00453-017-0322-4

    Article  MathSciNet  MATH  Google Scholar 

  10. Chimani, M., Gutwenger, C., Mutzel, P., Wolf, C.: Inserting a vertex into a planar graph. In: Proceedings of the 20th ACM-SIAM Symposium on Discrete Algorithms (SODA 2009), pp. 375–383. SIAM (2009). https://doi.org/10.1137/1.9781611973068.42

  11. Chimani, M., Hlinený, P.: Inserting multiple edges into a planar graph. In: Proceedings of the 32nd International Symposium on Computational Geometry (SoCG 2016), vol. 51, pp. 30:1–30:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016). https://doi.org/10.4230/LIPIcs.SoCG.2016.30

  12. Da Lozzo, G., Di Battista, G., Frati, F.: Extending upward planar graph drawings. In: Friggstad, Z., Sack, J.-R., Salavatipour, M.R. (eds.) WADS 2019. LNCS, vol. 11646, pp. 339–352. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-24766-9_25

    Chapter  Google Scholar 

  13. Felsner, S., Goodman, J.E.: Pseudoline arrangements. In: Tóth, C.D., O’Rourke, J., Goodman, J.E. (eds.) Handbook of Discrete and Computational Geometry, 3rd edn., pp. 125–157. CRC Press, Boca Raton. (2017)

    Google Scholar 

  14. Grünbaum, B.: Arrangements and spreads. AMS (1972)

    Google Scholar 

  15. Gutwenger, C., Mutzel, P., Weiskircher, R.: Inserting an edge into a planar graph. Algorithmica 41(4), 289–308 (2005). https://doi.org/10.1007/s00453-004-1128-8

    Article  MathSciNet  MATH  Google Scholar 

  16. Hajnal, P., Igamberdiev, A., Rote, G., Schulz, A.: Saturated simple and 2-simple topological graphs with few edges. J. Graph Algorithms Appl. 22(1), 117–138 (2018). https://doi.org/10.7155/jgaa.00460

    Article  MathSciNet  MATH  Google Scholar 

  17. Jelínek, V., Kratochvíl, J., Rutter, I.: A Kuratowski-type theorem for planarity of partially embedded graphs. Comput. Geom. Theory Appl. 46(4), 466–492 (2013). https://doi.org/10.1016/j.comgeo.2012.07.005. Special Issue on the 27th Annual Symposium on Computational Geometry (SoCG’11)

    Article  MathSciNet  MATH  Google Scholar 

  18. Klavík, P., Kratochvíl, J., Krawczyk, T., Walczak, B.: Extending partial representations of function graphs and permutation graphs. In: Epstein, L., Ferragina, P. (eds.) ESA 2012. LNCS, vol. 7501, pp. 671–682. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-33090-2_58

    Chapter  Google Scholar 

  19. Klavík, P., et al.: Extending partial representations of proper and unit interval graphs. Algorithmica 77(4), 1071–1104 (2016). https://doi.org/10.1007/s00453-016-0133-z

  20. Klavík, P., Kratochvíl, J., Otachi, Y., Saitoh, T.: Extending partial representations of subclasses of chordal graphs. Theor. Comput. Sci. 576, 85–101 (2015). https://doi.org/10.1016/j.tcs.2015.02.007

    Article  MathSciNet  MATH  Google Scholar 

  21. Klavík, P., Kratochvíl, J., Otachi, Y., Saitoh, T., Vyskočil, T.: Extending partial representations of interval graphs. Algorithmica 78(3), 945–967 (2016). https://doi.org/10.1007/s00453-016-0186-z

    Article  MathSciNet  MATH  Google Scholar 

  22. Kynčl, J., Pach, J., Radoičić, R., Tóth, G.: Saturated simple and \(k\)-simple topological graphs. Comput. Geom. Theory Appl. 48(4), 295–310 (2015). https://doi.org/10.1016/j.comgeo.2014.10.008

    Article  MathSciNet  MATH  Google Scholar 

  23. Kynčl, J.: Improved enumeration of simple topological graphs. Discrete Comput. Geom. 50(3), 727–770 (2013). https://doi.org/10.1007/s00454-013-9535-8

    Article  MathSciNet  MATH  Google Scholar 

  24. Levi, F.: Die Teilung der projektiven Ebene durch Gerade oder Pseudogerade. Berichte über die Verhandlungen der Sächsischen Akademie der Wissenschaften zu Leipzig, Mathematisch-Physische Klasse 78, 256–267 (1926). (in German)

    MATH  Google Scholar 

  25. Mchedlidze, T., Nöllenburg, M., Rutter, I.: Extending convex partial drawings of graphs. Algorithmica 76(1), 47–67 (2015). https://doi.org/10.1007/s00453-015-0018-6

    Article  MathSciNet  MATH  Google Scholar 

  26. Pach, J., Brass, P., Moser, W.O.J.: Research Problems in Discrete Geometry. Springer, New York (2005). https://doi.org/10.1007/0-387-29929-7

    Book  MATH  Google Scholar 

  27. Patrignani, M.: On extending a partial straight-line drawing. Int. J. Found. Comput. Sci. 17(5), 1061–1070 (2006). https://doi.org/10.1142/S0129054106004261

    Article  MathSciNet  MATH  Google Scholar 

  28. Radermacher, M., Rutter, I.: Inserting an edge into a geometric embedding. In: Biedl, T., Kerren, A. (eds.) GD 2018. LNCS, vol. 11282, pp. 402–415. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-04414-5_29

    Chapter  Google Scholar 

  29. Riskin, A.: The crossing number of a cubic plane polyhedral map plus an edge. Studia Scientiarum Mathematicarum Hungarica 31(4), 405–414 (1996)

    MathSciNet  MATH  Google Scholar 

  30. Schaefer, M.: Crossing Numbers of Graphs. CRC Press, Boca Raton (2018)

    Book  Google Scholar 

  31. Schaefer, M.: A proof of Levi’s extension lemma. ArXiv e-Prints (2019)

    Google Scholar 

  32. Snoeyink, J., Hershberger, J.: Sweeping arrangements of curves. In: Discrete and Computational Geometry: Papers from the DIMACS Special Year, vol. 6, pp. 309–350. DIMACS/AMS (1991). https://doi.org/10.1090/dimacs/006/21

  33. Sturmfels, B., Ziegler, G.M.: Extension spaces of oriented matroids. Discrete Comput. Geom. 10(1), 23–45 (1993). https://doi.org/10.1007/BF02573961

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was started during the 6th Austrian-Japanese-Mexican-Spanish Workshop on Discrete Geometry in Juni 2019 in Austria. We thank all the participants for the good atmosphere as well as discussions on the topic. Also, we thank Jan Kynčl for sending us remarks on the first arXiv version of this work and an anonymous referee for further helpful comments.

Author information

Authors and Affiliations

  1. IST Austria, Klosterneuburg, Austria

    Alan Arroyo

  2. Utrecht University, Utrecht, The Netherlands

    Fabian Klute

  3. TU Eindhoven, Eindhoven, The Netherlands

    Irene Parada

  4. Universität des Saarlandes, Saarbrücken, Germany

    Raimund Seidel

  5. Graz University of Technology, Graz, Austria

    Birgit Vogtenhuber

  6. Osnabrück University, Osnabrück, Germany

    Tilo Wiedera

Authors
  1. Alan Arroyo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fabian Klute
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Irene Parada
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Raimund Seidel
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Birgit Vogtenhuber
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Tilo Wiedera
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Birgit Vogtenhuber .

Editor information

Editors and Affiliations

  1. University of Leeds, Leeds, UK

    Isolde Adler

  2. University of Leeds, Leeds, UK

    Haiko Müller

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Arroyo, A., Klute, F., Parada, I., Seidel, R., Vogtenhuber, B., Wiedera, T. (2020). Inserting One Edge into a Simple Drawing Is Hard. In: Adler, I., Müller, H. (eds) Graph-Theoretic Concepts in Computer Science. WG 2020. Lecture Notes in Computer Science(), vol 12301. Springer, Cham. https://doi.org/10.1007/978-3-030-60440-0_26

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-60440-0_26

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-60439-4

  • Online ISBN: 978-3-030-60440-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics