Spatial graphs with local knots (Q365136)

In 1987, \textit{S. Suzuki} [``A prime decomposition theorem for a graph in the 3-sphere'', in: Topology and computer science, (Atami, 1986), 259--276, Kinokuniya, Tokyo, (1987)] extended a classic result of \textit{H. Schubert} [Sitzungsber. Heidelberger Akad. Wiss., Math.-Naturw. Kl. 1949, No.3, 50 S. (1949; Zbl 0031.28602)] to spatial graphs by providing a prime decomposition theorem for graphs in \(S^3\). This paper continues in this vein and shows that every locally knotted edge of a 3-connected spatial graph has a ball meeting the graph in an arc that contains all the local knots of that edge. It is shown that this unknotting ball is unique up to an isotopy that setwise fixes the graph. This result is applied to the study of topological symmetry groups that preserve orientation of certain graphs in \(S^3\).