Finding the longest path between two nodes in a bidirectional unweighted graph
Williams (2009) [arXiv version] gives a randomized $2^k poly(n)$ time algorithm finding a path of length at least $k$ in a graph on $n$ vertices. The paper contains pointers to previous deterministic and randomized algorithms. Many of these algorithms can probably be modified to find a $k$-path whose endpoints are given in the input.