• DOI:10.1007/978-3-540-70575-8_18
• Corpus ID: 8530563

Treewidth computation and extremal combinatorics

@article{Fomin2008TreewidthCA,
  title={Treewidth computation and extremal combinatorics},
  author={F. Fomin and Yngve Villanger},
  journal={Combinatorica},
  year={2008},
  volume={32},
  pages={289-308}
}

• F. Fomin, Yngve Villanger
• Published 9 March 2008
• Mathematics
• Combinatorica

AbstractFor a given graph G and integers b,f ≥0, let S be a subset of vertices of G of size b+1 such that the subgraph of G induced by S is connected and S can be separated from other vertices of G by removing f vertices. We prove that every graph on n vertices contains at most $n\left( {_b^{b + f} } \right)$ such vertex subsets. This result from extremal combinatorics appears to be very useful in the design of several enumeration and exact algorithms. In particular, we use it to provide algorithms that for a given n-vertex graph G•compute the treewidth of G in time O(1.7549n) by making use of exponential space and in time O(2.6151n) and polynomial space•decide in time O(n5·$_{k + 2}^{\left\lceil {(2n + k + 8)/3} \right\rceil } $) if the treewidth of G is at most k•list all minimal separators of G in time O(1.6181n) and all potential maximal cliques of G in time O(1.7549n).
This significantly improves previous algorithms for these problems.