Say we are solving a tree decomposition problem, e.g. given a graph $G = (V, E)$ we try to find a chordal graph $H$ such that $V(H) = V(G)$, $E(G) \in E(H)$ and the maximal clique in $H$ is minimal among all possible $H$.
This problem is known to be NP-hard. For other/better definitions of tree decomposition/treewidth see Wikipedia
Is there a way to reduce treewidth to some other NP problems, especially Maximum Independent Set/Maximum clique/SAT or similar ones?
I found a lot of results of sort "if a graph has bounded treewidth, then problem XXX has a polynomial solution", but nothing like "if we can solve XXX, then we can find the tree decomposition"
