TY - JOUR
T1 - Linkless embeddings of graphs in 3-space
AU - Robertson, Neil
AU - Seymour, P. D.
AU - Thomas, Robin
N1 - Copyright:
Copyright 2016 Elsevier B.V., All rights reserved.
PY - 1993/1
Y1 - 1993/1
N2 - 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 K5 or K3,3. (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 K6 by means of Y∆- and ∆Y-exchanges.
AB - 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 K5 or K3,3. (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 K6 by means of Y∆- and ∆Y-exchanges.
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U2 - 10.1090/S0273-0979-1993-00335-5
DO - 10.1090/S0273-0979-1993-00335-5
M3 - Comment/debate
AN - SCOPUS:0011409169
VL - 28
SP - 84
EP - 89
JO - Bulletin of the American Mathematical Society
JF - Bulletin of the American Mathematical Society
SN - 0273-0979
IS - 1
ER -