  Maximum throughput requires path diversity enabled by bifurcating traffic at
different network nodes. In this work, we consider a network where traffic
bifurcation is allowed only at a subset of nodes called \emph{routers}, while
the rest nodes (called \emph{forwarders}) cannot bifurcate traffic and hence
only forward packets on specified paths. This implements an overlay network of
routers where each overlay link corresponds to a path in the physical network.
We study dynamic routing implemented at the overlay. We develop a queue-based
policy, which is shown to be maximally stable (throughput optimal) for a
restricted class of network scenarios where overlay links do not correspond to
overlapping physical paths. Simulation results show that our policy yields
better delay over dynamic policies that allow bifurcation at all nodes, such as
the backpressure policy. Additionally, we provide a heuristic extension of our
proposed overlay routing scheme for the unrestricted class of networks.
