Linkless embeddings of graphs in space
Authors:
Neil Robertson, P. D. Seymour and Robin Thomas
Journal:
Bull. Amer. Math. Soc. 28 (1993), 8489
MSC:
Primary 57M15; Secondary 05C10
MathSciNet review:
1164063
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Abstract: We announce results about flat (linkless) embeddings of graphs in 3space. A piecewiselinear embedding of a graph in 3space 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 3space 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 "3switches", an analog of 2switches from the theory of planar embeddings. In particular, any two flat embeddings of a 4connected 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:
http://dx.doi.org/10.1090/S027309791993003355
PII:
S 02730979(1993)003355
Article copyright:
© Copyright 1993
American Mathematical Society
