Symmetrization theorem of full Steiner trees (Q1328395)

The paper contributes to the plane Steiner problem by a symmetrization theorem. The topology of a full Steiner tree with an even number of regular points (and of Steiner points) is called symmetric iff there is a fixpoint-free 1-1 correspondence of all points preserving adjacency and preserving the circular ordering of the neighbors of each Steiner point. It is shown that to each full Steiner tree on \(2n\) (regular) points with symmetric topology there exists a full Steiner tree on a central symmetric set of \(2n\) points having the same length. This result is used to give a short proof for a result of \textit{H. O. Pollak} on quadrilaterals, see [J. Comb. Theory, Ser. A 24, 278-295 (1978; Zbl 0392.05021)].