  The VPN Tree Routing Conjecture is a conjecture about the Virtual Private
Network Design problem. It states that the symmetric version of the problem
always has an optimum solution which has a tree-like structure. In recent work,
Hurkens, Keijsper and Stougie (Proc. IPCO XI, 2005; SIAM J. Discrete Math.,
2007) have shown that the conjecture holds when the network is a ring. A
shorter proof of the VPN Conjecture for rings was found a few months ago by
Grandoni, Kaibel, Oriolo and Skutella (to appear in Oper. Res. Lett., 2008). In
their paper, Grandoni et al. introduce another conjecture, called the Pyramidal
Routing Conjecture (or simply PR Conjecture), which implies the VPN Conjecture.
Here we consider a strengthened version of the PR Conjecture. First we
establish several general tools which can be applied in arbitrary networks.
Then we use them to prove that outerplanar networks satisfy the PR Conjecture.
