Minimum Steiner Trees on a Set of Concyclic Points and Their Center

Consider a configuration of points comprising a point q and a set of concyclic points R that are all a given distance r from q in the Euclidean plane. In this paper, we investigate the relationship between the length of a minimum Steiner tree (MStT) on R?{q} and a minimum spanning tree on R. We show that if the degree of q in the MStT is 1, then the difference between these two lengths is at least (2-3)r, and that this lower bound is tight. This bound can be applied as part of an efficient algorithm to find the solution to the prize-collecting Euclidean Steiner tree problem, as outlined in an earlier paper.