Advertisement

Picking Planar Edges; or, Drawing a Graph with a Planar Subgraph

  • Marcus Schaefer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8871)

Abstract

Given a graph G and a subset F ⊆ E(G) of its edges, is there a drawing of G in which all edges of F are free of crossings? We show that this question can be solved in polynomial time using a Hanani-Tutte style approach. If we require the drawing of G to be straight-line, but allow up to one crossing along each edge in F, the problem turns out to be as hard as the existential theory of the real numbers.

Keywords

Dual Graph Existential Theory Line Arrangement Partial Planarity Embed Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Angelini, P., Binucci, C., Da Lozzo, G., Didimo, W., Grilli, L., Montecchiani, F., Patrignani, M., Tollis, I.G.: Drawings of non-planar graphs with crossing-free subgraphs. In: Wismath, S., Wolff, A. (eds.) GD 2013. LNCS, vol. 8242, pp. 292–303. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  2. 2.
    Angelini, P., Di Battista, G., Frati, F., Jelínek, V., Kratochvíl, J., Patrignani, M., Rutter, I.: Testing planarity of partially embedded graphs. In: Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, pp. 202–221. Society for Industrial and Applied Mathematics, Philadelphia (2010)CrossRefGoogle Scholar
  3. 3.
    Angelini, P., Da Lozzo, G., Neuwirth, D.: On some \(\mathcal{NP}\)-complete SEFE problems. In: Pal, S.P., Sadakane, K. (eds.) WALCOM 2014. LNCS, vol. 8344, pp. 200–212. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  4. 4.
    Cabello, S., Mohar, B.: Adding one edge to planar graphs makes crossing number and 1-planarity hard. SIAM Journal on Computing 42(5), 1803–1829 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Canny, J.: Some algebraic and geometric computations in pspace. In: STOC 1988: Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing, pp. 460–469. ACM, New York (1988)CrossRefGoogle Scholar
  6. 6.
    Chojnacki, C. (Hanani, H.): Über wesentlich unplättbare Kurven im drei-dimensionalen Raume. Fundamenta Mathematicae 23, 135–142 (1934)Google Scholar
  7. 7.
    de Longueville, M.: A course in topological combinatorics. Universitext. Springer, New York (2013)CrossRefzbMATHGoogle Scholar
  8. 8.
    Eggleton, R.B.: Rectilinear drawings of graphs. Utilitas Math. 29, 149–172 (1986)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Eppstein, D.: Big batch of graph drawing preprints, http://11011110.livejournal.com/275238.html (last accessed September 4, 2013)
  10. 10.
    Grigoriev, A., Bodlaender, H.L.: Algorithms for graphs embeddable with few crossings per edge. Algorithmica 49(1), 1–11 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Gutwenger, C., Mutzel, P., Schaefer, M.: Practical experience with Hanani-Tutte for testing c-planarity. In: McGeoch, C.C., Meyer, U. (eds.) 2014 Proceedings of the Sixteenth Workshop on Algorithm Engineering and Experiments (ALENEX), pp. 86–97. SIAM (2014)Google Scholar
  12. 12.
    Hong, S.-H., Eades, P., Liotta, G., Poon, S.-H.: Fáry’s theorem for 1-planar graphs. In: Gudmundsson, J., Mestre, J., Viglas, T. (eds.) COCOON 2012. LNCS, vol. 7434, pp. 335–346. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  13. 13.
    Jelínek, V., Kratochvíl, J., Rutter, I.: A Kuratowski-type theorem for planarity of partially embedded graphs. Comput. Geom. 46(4), 466–492 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Korzhik, V.P., Mohar, B.: Minimal obstructions for 1-immersions and hardness of 1-planarity testing. In: Tollis, I.G., Patrignani, M. (eds.) GD 2008. LNCS, vol. 5417, pp. 302–312. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  15. 15.
    Kratochvíl, J.: String graphs. I. The number of critical nonstring graphs is infinite. J. Combin. Theory Ser. B 52(1), 53–66 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Kratochvíl, J.: Crossing number of abstract topological graphs. In: Whitesides, S.H. (ed.) GD 1998. LNCS, vol. 1547, pp. 238–245. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  17. 17.
    Mäkinen, E.: On circular layouts. International Journal of Computer Mathematics 24(1), 29–37 (1988)CrossRefzbMATHGoogle Scholar
  18. 18.
    Nagamochi, H.: Straight-line drawability of embedded graphs. Technical Report 2013-005, Kyoto University (2013)Google Scholar
  19. 19.
    Pelsmajer, M.J., Schaefer, M., Štefankovič, D.: Removing independently even crossings. SIAM Journal on Discrete Mathematics 24(2), 379–393 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Schaefer, M.: Complexity of some geometric and topological problems. In: Eppstein, D., Gansner, E.R. (eds.) GD 2009. LNCS, vol. 5849, pp. 334–344. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  21. 21.
    Schaefer, M.: Toward a theory of planarity: Hanani-Tutte and planarity variants. Journal of Graph Algortihms and Applications 17(4), 367–440 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Schaefer, M.: Hanani-Tutte and related results. In: Bárány, I., Böröczky, K.J., Fejes Tóth, G., Pach, J. (eds.) Geometry—Intuitive, Discrete, and Convex—A Tribute to László Fejes Tóth. Bolyai Society Mathematical Studies, vol. 24. Springer, Berlin (2014)Google Scholar
  23. 23.
    Schaefer, M., Sedgwick, E., Štefankovič, D.: Recognizing string graphs in NP. In: Proceedings of the 33th Annual ACM Symposium on Theory of Computing (STOC 2002) (2002)Google Scholar
  24. 24.
    Schaefer, M., Štefankovič, D.: Fixed points, Nash equilibria, and the existential theory of the reals. Unpublished manuscript (2009)Google Scholar
  25. 25.
    Shor, P.W.: Stretchability of pseudolines is NP-hard. In: Applied Geometry and Discrete Mathematics. DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 4, pp. 531–554. Amer. Math. Soc., Providence (1991)Google Scholar
  26. 26.
    Thomassen, C.: Rectilinear drawings of graphs. J. Graph Theory 12(3), 335–341 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Tutte, W.T.: Toward a theory of crossing numbers. J. Combinatorial Theory 8, 45–53 (1970)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Wikipedia. Existential theory of the reals (2012), http://en.wikipedia.org/wiki/Existential_theory_of_the_reals (Online; accessed July 17, 2013)

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Marcus Schaefer
    • 1
  1. 1.School of ComputingDePaul UniversityChicagoUSA

Personalised recommendations