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Linkless embeddings of graphs in  -space
Author(s):
Neil
Robertson;
P. D.
Seymour;
Robin
Thomas
Journal:
Bull. Amer. Math. Soc.
28
(1993),
84-89.
MathSciNet review:
1164063
Retrieve article in:
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Abstract |
References |
Additional information
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.
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Additional Information:
DOI:
10.1090/S0273-0979-1993-00335-5
PII:
S 0273-0979(1993)00335-5
Copyright of article:
Copyright
1993,
American Mathematical Society
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