  Potential maximal cliques and minimal separators are combinatorial objects
which were introduced and studied in the realm of minimal triangulations
problems including Minimum Fill-in and Treewidth. We discover unexpected
applications of these notions to the field of moderate exponential algorithms.
In particular, we show that given an n-vertex graph G together with its set of
potential maximal cliques Pi_G, and an integer t, it is possible in time |Pi_G|
* n^(O(t)) to find a maximum induced subgraph of treewidth t in G; and for a
given graph F of treewidth t, to decide if G contains an induced subgraph
isomorphic to F. Combined with an improved algorithm enumerating all potential
maximal cliques in time O(1.734601^n), this yields that both problems are
solvable in time 1.734601^n * n^(O(t)).
