On planarity of graphs in 3-manifolds
The author defines a graph \(\Gamma\) in a 3-manifold \(M\) to be planar (I would prefer to say planar in \(M\)) if \(\Gamma\) lies in a 2-sphere in \(M\) and then studies the problem of when an abstractly planar graph in a 3- manifold \(M\) is planar (in \(M\)). One result: \(\Gamma\) is planar if and only if each cycle of \(\Gamma\) bounds a disk in \(M\) whose interior misses \(\Gamma\). This is used to give another proof of a theorem of Scharlemann and Thompson that an abstractly planar graph \(\Gamma\) in \(S^ 3\) is planar if every proper subgraph is planar and \(\pi_ 1(S^ 3 - \Gamma)\) is a free group.
scientific article; zbMATH DE number 167712