Treewidth Computation and Extremal Combinatorics
نویسندگان
چکیده
For 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 ` b+f b ́ 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(1.7549) by making use of exponential space and in time O(2.6151) and polynomial space; – decide in time O(( 2n+k+1 3 ) · kn) if the treewidth of G is at most k; – list all minimal separators of G in time O(1.6181) and all potential maximal cliques of G in time O(1.7549). This significantly improves previous algorithms for these problems.
منابع مشابه
Essays in extremal combinatorics
We prove several results from different areas of extremal combinatorics, giving complete or partial solutions to a number of open problems. These results, coming from areas such as extremal graph theory, Ramsey theory and additive combinatorics, have been collected together because in each case the relevant proofs are quite short.
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Short Proofs of Some Extremal Results
We prove several results from different areas of extremal combinatorics, giving complete or partial solutions to a number of open problems. These results, coming from areas such as extremal graph theory, Ramsey theory and additive combinatorics, have been collected together because in each case the relevant proofs are quite short.
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عنوان ژورنال:
- Combinatorica
دوره 32 شماره
صفحات -
تاریخ انتشار 2008