The Complexity of Drawing Graphs on Few Lines and Few Planes

  • Steven Chaplick
  • Krzysztof Fleszar
  • Fabian Lipp
  • Alexander Ravsky
  • Oleg Verbitsky
  • Alexander Wolff
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10389)

Abstract

It is well known that any graph admits a crossing-free straight-line drawing in \(\mathbb {R} ^3\) and that any planar graph admits the same even in \(\mathbb {R} ^2\). For a graph G and \(d \in \{2,3\}\), let \(\rho ^1_d(G)\) denote the minimum number of lines in \(\mathbb {R} ^d\) that together can cover all edges of a drawing of G. For \(d=2\), G must be planar. We investigate the complexity of computing these parameters and obtain the following hardness and algorithmic results.

  • For \(d\in \{2,3\}\), we prove that deciding whether \(\rho ^1_d(G)\le k\) for a given graph G and integer k is \(\exists \mathbb {R}\)-complete.

  • Since \(\mathrm {NP}\subseteq \exists \mathbb {R}\), deciding \(\rho ^1_d(G)\le k\) is NP-hard for \(d\in \{2,3\}\). On the positive side, we show that the problem is fixed-parameter tractable with respect to k.

  • Since \(\exists \mathbb {R}\subseteq \mathrm {PSPACE}\), both \(\rho ^1_2(G)\) and \(\rho ^1_3(G)\) are computable in polynomial space. On the negative side, we show that drawings that are optimal with respect to \(\rho ^1_2\) or \(\rho ^1_3\) sometimes require irrational coordinates.

  • Let \(\rho ^2_3(G)\) be the minimum number of planes in \(\mathbb {R} ^3\) needed to cover a straight-line drawing of a graph G. We prove that deciding whether \(\rho ^2_3(G)\le k\) is NP-hard for any fixed \(k \ge 2\). Hence, the problem is not fixed-parameter tractable with respect to k unless \(\mathrm {P}=\mathrm {NP}\).

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References

  1. 1.
    Bienstock, D.: Some provably hard crossing number problems. Discrete Comput. Geom. 6, 443–459 (1991)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bose, P., Everett, H., Wismath, S.K.: Properties of arrangement graphs. Int. J. Comput. Geom. Appl. 13(6), 447–462 (2003)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Canny, J.F.: Some algebraic and geometric computations in PSPACE. In: Proc. 20th Annu. ACM Symp. Theory Comput. (STOC 1988), pp. 460–467 (1988)Google Scholar
  4. 4.
    Cardinal, J.: Computational geometry column 62. SIGACT News 46(4), 69–78 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chaplick, S., Fleszar, K., Lipp, F., Ravsky, A., Verbitsky, O., Wolff, A.: The complexity of drawing graphs on few lines and few planes (2016). arxiv.org/1607.06444
  6. 6.
    Chaplick, S., Fleszar, K., Lipp, F., Ravsky, A., Verbitsky, O., Wolff, A.: Drawing graphs on few lines and few planes. In: Hu, Y., Nöllenburg, M. (eds.) GD 2016. LNCS, vol. 9801, pp. 166–180. Springer, Cham (2016). doi:10.1007/978-3-319-50106-2_14 CrossRefGoogle Scholar
  7. 7.
    Dujmović, V., Eppstein, D., Suderman, M., Wood, D.R.: Drawings of planar graphs with few slopes and segments. Comput. Geom. Theory Appl. 38(3), 194–212 (2007)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Dujmović, V., Pór, A., Wood, D.R.: Track layouts of graphs. Discrete Math. & Theor. Comput. Sci. 6(2), 497–522 (2004)MathSciNetMATHGoogle Scholar
  9. 9.
    Dujmović, V., Whitesides, S.: Three-dimensional drawings. In: Tamassia, R. (ed.) Handbook of Graph Drawing and Visualization, chap. 14, pp. 455–488. CRC Press (2013)Google Scholar
  10. 10.
    Durocher, S., Mondal, D., Nishat, R.I., Whitesides, S.: A note on minimum-segment drawings of planar graphs. J. Graph Alg. Appl. 17(3), 301–328 (2013)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Eppstein, D.: Drawing arrangement graphs in small grids, or how to play planarity. J. Graph Alg. Appl. 18(2), 211–231 (2014)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Farrugia, A.: Vertex-partitioning into fixed additive induced-hereditary properties is NP-hard. Electr. J. Comb. 11(1) (2004)Google Scholar
  13. 13.
    Kang, R.J., Müller, T.: Sphere and dot product representations of graphs. Discrete Comput. Geom. 47(3), 548–568 (2012)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Kratochvíl, J., Lubiw, A., Nešetřil, J.: Noncrossing subgraphs in topological layouts. SIAM J. Discrete Math. 4(2), 223–244 (1991)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Kratochvíl, J., Matoušek, J.: Intersection graphs of segments. J. Comb. Theory, Ser. B 62(2), 289–315 (1994)Google Scholar
  16. 16.
    Matoušek, J.: Intersection graphs of segments and \({\exists }\mathbb{R}\) (2014). arxiv.org/1406.2636
  17. 17.
    Mnëv, N.E.: On manifolds of combinatorial types of projective configurations and convex polyhedra. Sov. Math., Dokl. 32, 335–337 (1985)MATHGoogle Scholar
  18. 18.
    Mnëv, N.E.: The universality theorems on the classification problem of configuration varieties and convex polytopes varieties. In: Viro, O. (ed.) Topology and Geometry, Rohlin Seminar 1984–1986. LNM, vol. 1346, pp. 527–543. Springer, Heidelberg (1988)CrossRefGoogle Scholar
  19. 19.
    Mulzer, W., Rote, G.: Minimum-weight triangulation is NP-hard. J. ACM 55(2), 11 (2008)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Renegar, J.: On the computational complexity and geometry of the first-order theory of the reals, part I: Introduction. preliminaries. the geometry of semi-algebraic sets. the decision problem for the existential theory of the reals. J. Symb. Comput. 13(3), 255–300 (1992)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Renegar, J.: On the computational complexity and geometry of the first-order theory of the reals, part II: The general decision problem. preliminaries for quantifier elimination. J. Symb. Comput. 13(3), 301–328 (1992)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Renegar, J.: On the computational complexity and geometry of the first-order theory of the reals, part III: Quantifier elimination. J. Symb. Comput. 13(3), 329–352 (1992)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    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). doi:10.1007/978-3-642-11805-0_32 CrossRefGoogle Scholar
  24. 24.
    Schaefer, M., Štefankovič, D.: Fixed points, Nash equilibria, and the existential theory of the reals. Theory Comput. Syst. 1–22 (2015)Google Scholar
  25. 25.
    Shor, P.W.: Stretchability of pseudolines is NP-hard. In: Gritzmann, P., Sturmfels, B. (eds.) Applied Geometry and Discrete Mathematics-The Victor Klee Festschrift, DIMACS Series, vol. 4, pp. 531–554 (1991). Amer. Math. SocGoogle Scholar
  26. 26.
    Zuckerman, D.: Linear degree extractors and the inapproximability of max clique and chromatic number. Theory Comput. 3(1), 103–128 (2007)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Lehrstuhl für Informatik IUniversität WürzburgWürzburgGermany
  2. 2.Department of Mathematical EngineeringUniversidad de ChileSantiagoChile
  3. 3.Pidstryhach Institute for Applied Problems of Mechanics and MathematicsNational Academy of Sciences of UkraineLvivUkraine
  4. 4.Institut für InformatikHumboldt-Universität zu BerlinBerlinGermany

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