Bounding component sizes of two-connected Steiner networks

We consider the problem of constructing a shortest Euclidean 2-connected Steiner network in the plane (SMN) for a set of n terminals. This problem has natural applications in the design of survivable communication networks. In [P. Winter, M. Zachariasen, Two-connected Steiner networks: Structural properties, OR Letters 33 (2005) 395–402] we proved that all cycles in SMNs with Steiner points must have pairs of consecutive terminals of degree 2. We use this result and the notion of reduced block-bridge trees suggested by Luebke [E.L. Luebke, k-connected Steiner network problems, PhD thesis, University of North Carolina, USA, 2002] to show that no full Steiner tree in a SMN spans more than n/3 + 1 terminals.