Linkless embeddings of graphs in -space

Authors:
Neil Robertson, P. D. Seymour and Robin Thomas

Journal:
Bull. Amer. Math. Soc. **28** (1993), 84-89

MSC:
Primary 57M15; Secondary 05C10

DOI:
https://doi.org/10.1090/S0273-0979-1993-00335-5

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 or .

(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 by means of - and -exchanges.

**[1]**Thomas Böhme,*On spatial representations of graphs*, Contemporary methods in graph theory, Bibliographisches Inst., Mannheim, 1990, pp. 151–167. MR**1126225****[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), no. 4, 445–453. MR**722061**, https://doi.org/10.1002/jgt.3190070410**[4]**Gordon M. Fisher,*On the group of all homeomorphisms of a manifold*, Trans. Amer. Math. Soc.**97**(1960), 193–212. MR**0117712**, https://doi.org/10.1090/S0002-9947-1960-0117712-9**[5]**Dick Wick Hall,*A note on primitive skew curves*, Bull. Amer. Math. Soc.**49**(1943), 935–936. MR**0009442**, https://doi.org/10.1090/S0002-9904-1943-08065-2**[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**, https://doi.org/10.1090/S0002-9947-1969-0246276-2**[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]**Horst Sachs,*On a spatial analogue of Kuratowski’s theorem on planar graphs—an open problem*, Graph theory (Łagów, 1981) Lecture Notes in Math., vol. 1018, Springer, Berlin, 1983, pp. 230–241. MR**730653**, https://doi.org/10.1007/BFb0071633**[14]**H. Sachs,*On spatial representations of finite graphs*, Finite and infinite sets, Vol. I, II (Eger, 1981) Colloq. Math. Soc. János Bolyai, vol. 37, North-Holland, Amsterdam, 1984, pp. 649–662. MR**818267****[15]**H. Saran,*Constructive results in graph minors*:*Linkless embeddings*, Ph.D. thesis, University of California at Berkeley, 1989.**[16]**Martin Scharlemann and Abigail Thompson,*Detecting unknotted graphs in 3-space*, J. Differential Geom.**34**(1991), no. 2, 539–560. MR**1131443****[17]**Hassler Whitney,*2-Isomorphic Graphs*, Amer. J. Math.**55**(1933), no. 1-4, 245–254. MR**1506961**, https://doi.org/10.2307/2371127**[18]**Ying Qing Wu,*On planarity of graphs in 3-manifolds*, Comment. Math. Helv.**67**(1992), no. 4, 635–647. MR**1185812**, https://doi.org/10.1007/BF02566522

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DOI:
https://doi.org/10.1090/S0273-0979-1993-00335-5

Article copyright:
© Copyright 1993
American Mathematical Society