Finding next-to-shortest paths in a graph

We study the problem of finding the next-to-shortest paths in a graph. A next-to-shortest (u, v)-path is a shortest (u, v)-path amongst (u, v)paths with length strictly greater than the length of the shortest (u, v)path. In constrast to the situation in directed graphs, where the problem has been shown to be NP-hard, providing edges of length zero are allowed, we prove the somewhat surprising result that there is a polynomial time algorithm for the undirected version of the problem.