Treewidth computation and extremal combinatorics

For a given graph Gand integers b,f≥ 0, let Sbe a subset of vertices of Gofsize b+ 1 such that the subgraph of Ginduced bySis connected and Scan be separated from othervertices of Gby removing fvertices. We provethat every graph on nvertices contains at most$n\binom{b+f}{b}$ such vertex subsets. This result from extremalcombinatorics appears to be very useful in the design of severalenumeration and exact algorithms. In particular, we use it toprovide algorithms that for a given n-vertex graphG compute the treewidth of Gin time$\mathcal{O}(1.7549^n)$ by making use of exponential space and intime $\mathcal{O}(2.6151^n)$ and polynomial space; decide in time $\mathcal{O}(({\frac{2n+k+1}{3})^{k+1}\cdotkn^6})$ if the treewidth of Gis at most k; list all minimal separators of Gin time$\mathcal{O}(1.6181^n)$ and all potential maximal cliques ofGin time $\mathcal{O}(1.7549^n)$. This significantly improves previous algorithms for theseproblems.