Complexity of two cycles which differ by $1$ in length
It belongs to NP obviously, and is NP-complete by a reduction from the Hamilton cycle problem on bipartite graph.

Given a bipartite graph with $2n$ vertices, add an isolated simple cycle of length $2n+1$. Since there is no odd cycle in a bipartite graph, the new graph has two cycles which differ by 1 in length if and only if there is a Hamilton cycle in the bipartite graph.