The research reported in this paper deals with the problem of searching through an unknown terrain by a physical agent such as a robot. The unknown terrain over which the agent will travel is represented by an undirected graph. The agent has no prior knowledge of the graph. It can only learn about its environment by physically roaming it. Given a starting location s, the agent tries to reach a target location t using the minimum amount of physical movement. This problem, which is a natural generalization of the classical shortest path problem, will be referred to as the dynamic shortest path problem. Most of the classical shortest path algorithms perform very poorly in the scenario of a physical agent traversing an initially unknown search space. They do not attempt to minimize the amount of physical movement required by the agent to reach the goal location. In order to overcome the failings of these search algorithms in dealing with searches of this particular nature, a new search strategy, called persistence search, is developed and presented in this paper.