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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
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 $ {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.


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DOI: https://doi.org/10.1090/S0273-0979-1993-00335-5
Article copyright: © Copyright 1993 American Mathematical Society

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