Some structural and geometric properties of two-connected Steiner networks

Abstract

We consider the problem of constructing a shortest Euclidean 2-connected Steiner network (SMN) for a set of terminals. This problem has natural applica-tions in the design of survivable communication net-works. A SMN decomposes into components that are full Steiner trees. Winter and Zachariasen proved that all cycles in SMNs with Steiner points must have two pairs of consecutive terminals of degree 2. We use this result and the notion of reduced block-bridge trees of Luebke to show that no component in a SMN spans more than approximately one-third of the ter-minals. Furthermore, we show that no component spans more than two terminals on the boundary of the convex hull of the terminals; such two terminals must in addition be consecutive on the boundary of this convex hull. Algorithmic implications of these results are discussed. © 2007, Australian Computer Society, Inc