Two-connected Steiner networks: structural properties

We consider the problem of constructing a minimum-length 2-connected network for a set of
terminals in a graph where edge-weights satisfy the triangle inequality. As a special case of the
problem, we have the Euclidean 2-connected Steiner network problem in which a set of points in
the plane should be interconnected by a 2-connected network of minimum length. These problems
have natural applications in the design of surviable communication networks.
In this paper we give a number of structural results for the problem. We show that all cycles
must have at least four terminals, and that these terminals must be distributed in groups of two. Furthermore, we give a tight lower bound on the minimum size of any problem instance that requires
a Steiner vertex.