For fixed parameter $k$, I would like to find a long cycle of length $\geq k$ in an undirected graph $G(V,E)$. This can be done in $O(k!2^k|V|)$-time  using 1) depth-first search (DFS) and 2) dynamic programming on a bounded tree-width decomposition of $G$ derived from the DFS traversal.
However,  does not contain explicit details on what information should be stored in the tables associated with each bag of the tree decomposition.
When looking for long cycles using the dynamic programming method suggested by , are there any ideas or resources that specify what information needs to be stored at each bag and how to combine such tables?
 Gabow & Nie, 2008 contains an overview of the above problem, from the perspective of directed and undirected graphs, and includes a review of research on the undirected case.
 Bodlaender, 1993 suggests an $O(k!2^kn)$-time algorithm based on depth-first search (DFS) and dynamic programming.