Discrete & Computational Geometry

, Volume 47, Issue 4, pp 731–755 | Cite as

Linkless and Flat Embeddings in 3-Space

  • Ken-ichi Kawarabayashi
  • Stephan Kreutzer
  • Bojan Mohar
Article
  • 230 Downloads

Abstract

We consider piecewise linear embeddings of graphs in 3-space ℝ3. Such an embedding is linkless if every pair of disjoint cycles forms a trivial link (in the sense of knot theory). Robertson, Seymour and Thomas (J. Comb. Theory, Ser. B 64:185–227, 1995) showed that a graph has a linkless embedding in ℝ3 if and only if it does not contain as a minor any of seven graphs in Petersen’s family (graphs obtained from K 6 by a series of YΔ and ΔY operations). They also showed that a graph is linklessly embeddable in ℝ3 if and only if it admits a flat embedding into ℝ3, i.e. an embedding such that for every cycle C of G there exists a closed 2-disk D⊆ℝ3 with DG=∂D=C. Clearly, every flat embedding is linkless, but the converse is not true. We consider the following algorithmic problem associated with embeddings in ℝ3:

Flat Embedding: For a given graph G, either detect one of Petersen’s family graphs as a minor in G, or return a flat (and hence linkless) embedding of G in ℝ3.

The first outcome is a certificate that G has no linkless and no flat embeddings. Our main result is to give an O(n 2) algorithm for this problem. While there is a known polynomial-time algorithm for constructing linkless embeddings (van der Holst in J. Comb. Theory, Ser. B 99:512–530, 2009), this is the first polynomial-time algorithm for constructing flat embeddings in 3-space. This settles a problem proposed by Lovász (www.cs.elte.hu/~lovasz/klee.ppt, 2000).

Keywords

Linkless embedding Knot Unknot Flat embedding 

Notes

Acknowledgements

Part of this work was done while all the authors participated the workshop “Graph Minors” at BIRS (Banff International Research Station) in October 2008.

K.K. research was partly supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research, by C and C Foundation, by Kayamori Foundation and by Inoue Research Award for Young Scientists.

S.K. research was partly supported by DFG grant KR 2898/1-3.

B.M. was supported in part by an NSERC Discovery Grant, by the Canada Research Chair Program, and by the ARRS, Research Program P1-0297. On leave from Department of Mathematics, University of Ljubljana, Slovenia.

References

  1. 1.
    Arnborg, S., Lagergren, J., Seese, D.: Easy problems for tree-decomposable graphs. J. Algorithms 12(2), 308–340 (1991) MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Baker, B.S.: Approximation algorithms for NP-complete problems on planar graphs. J. ACM 41, 153–180 (1994) MATHCrossRefGoogle Scholar
  3. 3.
    Bodlaender, H.L.: A linear-time algorithm for finding tree-decomposition of small treewidth. SIAM J. Comput. 25, 1305–1317 (1996) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Chiba, N., Nishizeki, T., Abe, S., Ozawa, T.: A linear time algorithm for embedding planar graphs using PQ-trees. J. Comput. Syst. Sci. 30, 54–76 (1985) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Colin de Verdière, Y.: Sur un nouvel invariant des graphes et un critère de planarité. J. Comb. Theory, Ser. B 50, 11–21 (1990) MATHCrossRefGoogle Scholar
  6. 6.
    Colin de Verdière, Y.: Multiplicities of eigenvalues and tree-width of graphs. J. Comb. Theory, Ser. B 74, 121–146 (1998) MATHCrossRefGoogle Scholar
  7. 7.
    Colin de Verdière, Y.: On a new graph invariant and a criterion of planarity. In: Robertson, N., Seymour, P. (eds.) Graph Structure Theory. Contemp. Math. Amer. Math. Soc., vol. 147, pp. 137–147 (1993) CrossRefGoogle Scholar
  8. 8.
    Conway, J.H., Gordon, C.McA.: Knots and links in spatial graphs. J. Graph Theory 7, 445–453 (1983) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Cook, S.A., Reckhow, R.A.: Time bounded random access machines. J. Comput. Syst. Sci. 7, 354–375 (1976) MathSciNetCrossRefGoogle Scholar
  10. 10.
    Courcelle, B.: Graph rewriting: An algebraic and logic approach. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, vol. B, pp. 193–242. Elsevier, Amsterdam (1990) Google Scholar
  11. 11.
    Diestel, R.: Graph Theory, 4th edn. Springer, Berlin (2010) CrossRefGoogle Scholar
  12. 12.
    Diestel, R., Gorbunov, K.Yu., Jensen, T.R., Thomassen, C.: Highly connected sets and the excluded grid theorem. J. Comb. Theory, Ser. B 75, 61–73 (1999) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Grigni, M., Koutsoupias, E., Papadimitriou, C.: An approximation scheme for planar graphs TSP. In: Proc. 36th Symposium on Foundations of Computer Science (FOCS’95), pp. 640–646 (1995) Google Scholar
  14. 14.
    Hopcroft, J.E., Tarjan, R.: Efficient planarity testing. J. ACM 21, 549–568 (1974) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    van der Holst, H.: A polynomial-time algorithm to find a linkless embedding of a graph. J. Comb. Theory, Ser. B 99, 512–530 (2009) MATHCrossRefGoogle Scholar
  16. 16.
    Kawarabayashi, K.: Rooted minor problems in highly connected graphs. Discrete Math. 287, 121–123 (2004) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Kawarabayashi, K., Kreutzer, S., Mohar, B.: Linkless and flat embeddings in 3-space and the unknot problem. In: Symposium on Computational Geometry (SoCG), pp. 97–106 (2010) Google Scholar
  18. 18.
    Kawarabayashi, K., Mohar, B.: Graph and Map Isomorphism and all polyhedral embeddings in linear time. In: Proc. 40th ACM Symposium on Theory of Computing (STOC’08), pp. 471–480 (2008) Google Scholar
  19. 19.
    Kawarabayashi, K., Mohar, B., Reed, B.: A simpler linear time algorithm for embedding graphs on a surface and for bounded tree-width graphs. In: Proc. 49th Annual Symposium on Foundations of Computer Science (FOCS’08), pp. 771–780 (2008) CrossRefGoogle Scholar
  20. 20.
    Kawarabayashi, K., Kobayashi, Y., Reed, B.: The disjoint paths problem in quadratic time. J. Comb. Theory, Ser. B 102(2), 424–435 (2012) CrossRefGoogle Scholar
  21. 21.
    Kawarabayashi, K., Norine, S., Thomas, R., Wollan, P.: K 6 minors in large 6-connected graphs, submitted. Available at http://www.math.princeton.edu/~snorin/papers/k6large.pdf
  22. 22.
    Kawarabayashi, K., Reed, B.: Computing crossing number in linear time. In: Proc. 39th ACM Symposium on Theory of Computing (STOC’07), pp. 382–390 (2007) Google Scholar
  23. 23.
    Kawarabayashi, K., Wollan, P.: A shorter proof of the graph minor algorithm – the uniquely linkage theorem. In: Proc. 42nd ACM Symposium on Theory of Computing (STOC’10), pp. 687–694 (2010). Full version available at http://research.nii.ac.jp/~k_keniti/uniquelink.pdf CrossRefGoogle Scholar
  24. 24.
    Klein, P.N.: A linear time approximation scheme for TSP for planar weighted graphs. In: Proc. 46th Symposium on Foundations of Computer Science (FOCS’05), pp. 146–155 (2005) Google Scholar
  25. 25.
    Kuratowski, C.: Sur le problème des courbes gauches en topologie. Fundam. Math. 15, 271–283 (1930) MATHGoogle Scholar
  26. 26.
    van Leeuwen, J.: Graph algorithms. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science. Vol. A. Algorithms and Complexity, pp. 525–631. Elsevier/MIT Press, Amsterdam (1990) Google Scholar
  27. 27.
    Lipton, R.J., Tarjan, R.E.: Applications of a planar separator theorem. SIAM J. Comput. 9, 615–627 (1980) MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Lovász, L.: Steinitz representations (2000). http://www.cs.elte.hu/~lovasz/klee.ppt
  29. 29.
    Lovász, L., Schrijver, A.: On the null space of a Colin de Verdière matrix. Ann. Inst. Fourier (Grenoble) 49, 1017–1026 (1999) MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Lovász, L., Schrijver, A.: A Borsuk theorem for antipodal links and a spectral characterization of linklessly embeddable graphs. Proc. Am. Math. Soc. 126, 1275–1285 (1998) MATHCrossRefGoogle Scholar
  31. 31.
    Lucchesi, C.L., Younger, D.H.: A minimax theorem for directed graphs. J. Lond. Math. Soc. 17, 369–374 (1978) MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Mader, W.: Homomorphiesätze für Graphen. Math. Ann. 178, 154–168 (1968) MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Mohar, B.: Embedding graphs in an arbitrary surface in linear time. In: Proc. 28th ACM Symposium on Theory of Computing (STOC’96), pp. 392–397 (1996) Google Scholar
  34. 34.
    Mohar, B.: A linear time algorithm for embedding graphs in an arbitrary surface. SIAM J. Discrete Math. 12, 6–26 (1999) MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Mohar, B., Thomassen, C.: Graphs on Surfaces. Johns Hopkins University Press, Baltimore (2001) MATHGoogle Scholar
  36. 36.
    Motwani, R., Raghunathan, A., Saran, H.: Constructive results from graph minors: Linkless embeddings. In: Proc. 29th Annual Symposium on Foundations of Computer Science (FOCS’88), pp. 398–409 (1988) Google Scholar
  37. 37.
    Reed, B.: Tree width and tangles: a new connectivity measure and some applications. In: Surveys in Combinatorics, 1997 (London). London Math. Soc. Lecture Note Ser., vol. 241, pp. 87–162. Cambridge Univ. Press, Cambridge (1997) CrossRefGoogle Scholar
  38. 38.
    Reed, B., Wood:, D.: A linear time algorithm to find a separator in a graph with an excluded minor. ACM Trans. Algorithms 5(4) (2009) Google Scholar
  39. 39.
    Robertson, N., Seymour, P.D.: Graph minors. V. Excluding a planar graph. J. Comb. Theory, Ser. B 41, 92–114 (1986) MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Robertson, N., Seymour, P.D.: Graph Minors. IX. Disjoint crossed paths. J. Comb. Theory, Ser. B 49, 40–77 (1990) MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Robertson, N., Seymour, P.D.: Graph minors. XIII. The disjoint paths problem. J. Comb. Theory, Ser. B 63, 65–110 (1995) MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    Robertson, N., Seymour, P.D.: Graph minors. XX. Wagner’s Conjecture. J. Comb. Theory, Ser. B 92, 325–357 (2004) MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    Robertson, N., Seymour, P.D., Thomas, R.: Hadwiger’s conjecture for K6-free graphs. Combinatorica 13, 279–361 (1993) MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    Robertson, N., Seymour, P.D., Thomas, R.: Quickly excluding a planar graph. J. Comb. Theory, Ser. B 62, 323–348 (1994) MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    Robertson, N., Seymour, P.D., Thomas, R.: Kuratowski chains. J. Comb. Theory, Ser. B 64, 127–154 (1995) MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    Robertson, N., Seymour, P.D., Thomas, R.: Petersen family minors. J. Comb. Theory, Ser. B 64, 155–184 (1995) MathSciNetMATHCrossRefGoogle Scholar
  47. 47.
    Robertson, N., Seymour, P.D., Thomas, R.: Sachs’ linkless embedding conjecture. J. Comb. Theory, Ser. B 64, 185–227 (1995) MathSciNetMATHCrossRefGoogle Scholar
  48. 48.
    Robertson, N., Seymour, P.D., Thomas, R.: A survey of linkless embeddings. Contemp. Math. 147, 125–136 (1993) MathSciNetCrossRefGoogle Scholar
  49. 49.
    Seymour, P.: Disjoint paths in graphs. Discrete Math. 29, 293–309 (1980) MathSciNetMATHCrossRefGoogle Scholar
  50. 50.
    Seymour, P.D.: Packing circuits in eulerian digraphs. Combinatorica 16, 223–231 (1996) MathSciNetMATHCrossRefGoogle Scholar
  51. 51.
    Thomassen, C.: 2-linked graphs. Eur. J. Comb. 1, 371–378 (1980) MathSciNetMATHGoogle Scholar
  52. 52.
    Williamson, S.G.: Depth-first search and Kuratowski subgraphs. J. ACM 31, 681–693 (1984) MATHCrossRefGoogle Scholar
  53. 53.
    Wu, Y.-Q.: On planarity of graphs in 3-manifolds. Comment. Math. Helv. 67, 635–647 (1992) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Ken-ichi Kawarabayashi
    • 1
  • Stephan Kreutzer
    • 2
  • Bojan Mohar
    • 3
  1. 1.National Institute of InformaticsChiyoda-ku, TokyoJapan
  2. 2.Department of Computer ScienceUniversity of OxfordOxfordUK
  3. 3.Department of MathematicsSimon Fraser UniversityBurnabyCanada

Personalised recommendations