Simple path in a graph, within a given range of lengths
A special case of your problem is to determine whether there is a path of length between $n-1$ and $n-1$ (sic) in an $n$-vertex graph. This is the Hamiltonian path problem, which is well known to be NP-complete, so your problem is also NP-complete. More generally, it's also NP-complete to determine if a graph has a path of length at least&nbsp;$k$ (i.e., between $k$ and $n-1$) between two given vertices.