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Abstract
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A graph is
–apex
if it is planar after the deletion of at most two vertices. Such graphs are not
intrinsically knotted, IK. We investigate the converse, does not IK imply
–apex? We
determine the simplest possible counterexample, a graph on nine vertices and 21 edges that is neither
IK nor
–apex.
In the process, we show that every graph of 20 or fewer edges is
–apex.
This provides a new proof that an IK graph must have at least 21 edges. We also
classify IK graphs on nine vertices and 21 edges and find no new examples of minor
minimal IK graphs in this set.
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Keywords
spatial graph, intrinsic knotting, apex graph
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Mathematical Subject Classification 2000
Primary: 05C10
Secondary: 57M15
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Publication
Received: 29 October 2009
Accepted: 15 October 2010
Published: 11 March 2011
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