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Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)



Linkless embeddings of graphs in $ 3$-space

Authors: Neil Robertson, P. D. Seymour and Robin Thomas
Journal: Bull. Amer. Math. Soc. 28 (1993), 84-89
MSC: Primary 57M15; Secondary 05C10
MathSciNet review: 1164063
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Abstract: We announce results about flat (linkless) embeddings of graphs in 3-space. A piecewise-linear embedding of a graph in 3-space is called flat if every circuit of the graph bounds a disk disjoint from the rest of the graph. We have shown:

(i) An embedding is flat if and only if the fundamental group of the complement in 3-space of the embedding of every subgraph is free.

(ii) If two flat embeddings of the same graph are not ambient isotopic, then they differ on a subdivision of $ {K_5}$ or $ {K_{3,3}}$.

(iii) Any flat embedding of a graph can be transformed to any other flat embedding of the same graph by "3-switches", an analog of 2-switches from the theory of planar embeddings. In particular, any two flat embeddings of a 4-connected graph are either ambient isotopic, or one is ambient isotopic to a mirror image of the other.

(iv) A graph has a flat embedding if and only if it has no minor isomorphic to one of seven specified graphs. These are the graphs that can be obtained from $ {K_6}$ by means of $ Y \Delta$- and $ \Delta Y$-exchanges.

References [Enhancements On Off] (What's this?)

  • [1] T. Böhme, On spatial representations of graphs, Contemporary Methods in Graph Theory (R. Bodendieck, ed.), Mannheim, Wien, Zurich, 1990, pp. 151-167. MR 1126225 (93a:05053)
  • [2] -, Lecture at the AMS Summer Research Conference on Graph Minors, Seattle, WA, June 1991.
  • [3] J. H. Conway and C. McA. Gordon, Knots and links in spatial graphs, J. Graph Theory 7 (1983), 445-453. MR 722061 (85d:57002)
  • [4] G. M. Fisher, On the group of all homeomorphisms of a manifold, Trans. Amer. Math. Soc. 97 (1960), 193-212. MR 0117712 (22:8487)
  • [5] D. W. Hall, A note on primitive skew curves, Bull. Amer. Math. Soc. 49 (1943), 935-937. MR 0009442 (5:151b)
  • [6] C. Kuratowski, Sur le problème des courbes gauches en topologie, Fund. Math. 15 (1930), 271-283.
  • [7] W. K. Mason, Homeomorphic continuous curves in 2-space are isotopic in 3-space, Trans. Amer. Math. Soc. 142 (1969), 269-290. MR 0246276 (39:7580)
  • [8] R. Motwani, A. Raghunathan, and H. Saran, Constructive results from graph minors: Linkless embeddings, Proc. 29th Symposium on the Foundations of Computer Science, Yorktown Heights, 1988.
  • [9] N. Robertson and P. D. Seymour, Graph minors. XIII. The disjoint paths problem, submitted.
  • [10] N. Robertson, P. D. Seymour, and R. Thomas, Kuratowski chains, submitted.
  • [11] -, Petersen family minors, submitted.
  • [12] -, Sachs' linkless embedding conjecture, manuscript.
  • [13] H. Sachs, On spatial representation of finite graphs (Proceedings of a conference held in Lagów, February 10-13, 1981, Poland), Lecture Notes in Math., vol. 1018, Springer-Verlag, Berlin, Heidelberg, New York, and Tokyo, 1983. MR 730653 (85b:05077)
  • [14] -, On spatial representations of finite graphs, finite and infinite sets, (A. Hajnal, L. Lovász, and V. T. Sós, eds), Colloq. Math. Soc. János Bolyai, vol. 37, North-Holland, Budapest, 1984, pp. 649-662. MR 818267 (87c:05055)
  • [15] H. Saran, Constructive results in graph minors: Linkless embeddings, Ph.D. thesis, University of California at Berkeley, 1989.
  • [16] M. Scharlemann and A. Thompson, Detecting unknotted graphs in 3-space, J. Differential Geom. 34 (1991), 539-560. MR 1131443 (93a:57012)
  • [17] H. Whitney, 2-isomorphic graphs, Amer. J. Math. 55 (1933), 245-254. MR 1506961
  • [18] Y.-Q. Wu, On planarity of graphs in 3-manifolds, Comment. Math. Helv. (to appear). MR 1185812 (93m:57002)

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