Shortest paths between shortest paths

We study the following problem on reconfiguring shortest paths in graphs: Given two shortest s–t paths, what is the minimum number of steps required to transform one into the other, where each intermediate path must also be a shortest s–t path and must differ from the previous one by only one vertex. We prove that the shortest reconfiguration sequence can be exponential in the size of the graph and that it is NP-hard to compute the shortest reconfiguration sequence even when we know that the sequence has polynomial length.