The focus of this research is to develop minimum-cost dynamic routing policies that can identify connecting paths between nodes in a stochastic-state network. In this context, the stochastic element of the network is the network structure, that is, the set of links that exist under each realization of the network state. It is assumed that information about the true network state can be gathered only endogenously through the routing decisions themselves. As such, the objective becomes finding a dynamic policy that accounts for information gathered en route that minimizes the cost of detection of a viable path between a given origin and destination. An exact solution method, based on a Markovian decision process, is presented, and then a heuristic based on an aggregating function of the network is developed.