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


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).


Linkless embedding Knot Unknot Flat embedding 



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.


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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

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