2008
DOI: 10.1007/978-3-540-70575-8_18
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Treewidth Computation and Extremal Combinatorics

Abstract: Abstract. 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 bś uch 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 gr… Show more

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Cited by 36 publications
(49 citation statements)
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“…The following lemma of Fomin and Villanger [12] will be useful for us to guess connected sets of vertices with small running-time overhead. Along with Proposition 3 we use the following standard bound on binomial coefficients in terms of entropy.…”
Section: Proposition 2 Letmentioning
confidence: 99%
“…The following lemma of Fomin and Villanger [12] will be useful for us to guess connected sets of vertices with small running-time overhead. Along with Proposition 3 we use the following standard bound on binomial coefficients in terms of entropy.…”
Section: Proposition 2 Letmentioning
confidence: 99%
“…Let R be a minimum vertex cover of (V, F ), with k = |R|. We want to answer the question whether there exists an edge set 7549 n ) are the best known upper bounds for respectively the number of minimal separators and the number of potential maximal cliques [14]. Here, we will show that the restrictions given by F imply enough structure to bound the number of useful minimal separators and potential maximal cliques by a smaller function.…”
Section: Proposition 4 For a Potential Maximal Clique ω Of The Graphmentioning
confidence: 99%
“…It arises in particular in sparse matrix computations [16] and in perfect phylogeny since it has the problem of triangulating colored graphs as a special case [1,2]. It can also be seen as a generalization of the problems of adding or deleting edges in a minimum or minimal way in an arbitrary input graph to obtain a chordal graph, which have attracted considerable attention [13,14,19,20,23,24,26,27]. The NP-completeness of the problem follows from the results of several papers [1,6,32].…”
Section: Introductionmentioning
confidence: 99%
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“…For Treewidth there are several results: Arnborg et al give an O(n tw+2 ) time algorithm for treewidth, where tw is the treewidth of the graph [1]; Fomin et al give O(c n ) time algorithms with c < 2 [9,10]. These results are based on the property that the treewidth problem decomposes into independent subproblems if one bag or separator of the tree decomposition under construction is known.…”
Section: Introductionmentioning
confidence: 99%