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Detecting unknotted graphs in 3-space. (English) Zbl 0751.05033
The authors call a finite graph \(G\) abstractly planar if it is embeddable in \(S^ 2\) (or equivalently, in the plane), and planar if it lies on an embedded surface in \(S^ 3\) which is homeomorphic to \(S^ 2\). The main result is a necessary and sufficient condition for a graph in \(S^ 3\) to be planar: \(G\) is planar if and only if it is abstractly planar and for every subgraph \(G'\subseteq G\) the fundamental group \(\pi(S^ 3-G')\) is free.
This result can be viewed as an unknotting theorem in the style of Papakyriakopoulos: a simple closed curve (i.e., a graph with one vertex and one edge) in \(S^ 3\) is unknotted if and only if its complement in \(S^ 3\) has free fundamental group. The above theorem generalizes also a more recent result of C. McA. Gordon [Topology Appl. 27, 285-299 (1987; Zbl 0634.57007)], where \(G\) has a single vertex and an arbitrary number of edges.

05C10 Planar graphs; geometric and topological aspects of graph theory
57M25 Knots and links in the \(3\)-sphere (MSC2010)
05C75 Structural characterization of families of graphs
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