# Treewidth problem equivalence

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"