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### Listing all potential maximal cliques of a graph

Vincent Bouchitté, Ioan Todinca
2002 Theoretical Computer Science
A potential maximal clique of a graph is a vertex set that induces a maximal clique in some minimal triangulation of that graph.  ...  We show here that the potential maximal cliques of a graph can be generated in polynomial time in the number of minimal separators of the graph.  ...  The potential maximal cliques of a graph can be listed in polynomial time in its size and the number of its minimal separators.  ...

### Listing All Potential Maximal Cliques of a Graph [chapter]

Vincent Bouchitté, Ioan Todinca
2000 Lecture Notes in Computer Science
A potential maximal clique of a graph is a vertex set that induces a maximal clique in some minimal triangulation of that graph.  ...  We show here that the potential maximal cliques of a graph can be generated in polynomial time in the number of minimal separators of the graph.  ...  For all the classes mentioned above we can list the maximal sets of neighbor separators in polynomial time, so we uniÿed all the previous algorithms.  ...

### Treewidth Computation and Extremal Combinatorics [chapter]

Fedor V. Fomin, Yngve Villanger
2008 Lecture Notes in Computer Science
space; -decide in time O(( 2n+k+1 3 ) k+1 · n 6 ) if the treewidth of G is at most k; -list all minimal separators of G in time O(1.6181 n ) and all potential maximal cliques of G in time O(1.7549 n )  ...  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  ...  By Proposition 5, the treewidth of a graph can be computed in O(n 3 (|Π G | + |∆ G |)) time if the list of all minimal separators ∆ G and the list of all potential maximal cliques Π G for the graph is  ...

### Treewidth computation and extremal combinatorics

Fedor V. Fomin, Yngve Villanger
2012 Combinatorica
space; • decide in time O(n 5 · (2n+k+8)/3 k+2 ) if the treewidth of G is at most k; • list all minimal separators of G in time O(1.6181 n ) and all potential maximal cliques of G in time O(1.7549 n )  ...  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  ...  We are grateful to Saket Saurabh for many useful comments, and to the anonymous referee pointing out that one of the bounds matched the golden ratio.  ...

### Improved Exponential-Time Algorithms for Treewidth and Minimum Fill-In [chapter]

Yngve Villanger
2006 Lecture Notes in Computer Science
We show that the number of potential maximal cliques for an arbitrary graph G on n vertices is O * (1.8135 n ), and that all potential maximal cliques can be listed in O * (1.8899 n ) time.  ...  Their algorithm uses the notion of potential maximal cliques, and is able to list these in time O * (1.9601 n ), which gives the running time for the above mentioned problems.  ...  Theorem 9 . 9 A graph G on n vertices has at most 2n 3 nn ) potential maximal cliques. There is an algorithm to list all potential maximal cliques of a graph in time O * (1.8899 n ).Proof.  ...

### Treewidth computation and extremal combinatorics [article]

Fedor V. Fomin, Yngve Villanger
2008 arXiv   pre-print
space; - decide in time O((2n+k+1/3)^k+1· kn^6) if the treewidth of G is at most k; - list all minimal separators of G in time O(1.6181^n) and all potential maximal cliques of G in time O(1.7549^n).  ...  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  ...  We are grateful to Saket Saurabh for many useful comments, and to the anonymous referee pointing out that one of the bounds matched the golden ratio.  ...

### Exact Algorithms for Treewidth and Minimum Fill-In

Fedor V. Fomin, Dieter Kratsch, Ioan Todinca, Yngve Villanger
2008 SIAM journal on computing (Print)
Our result is based on a combinatorial proof that each graph on n vertices has at most n · 1.7087 n minimal separators and that all potential maximal cliques can be listed in O(1.8899 n ) time.  ...  For the class of AT-free graphs we obtain O(1.4142 n ) time algorithms to compute treewidth and minimum fill-in.  ...  There is an algorithm to list all potential maximal cliques of a graph in time O * (1.8899 n ).Proof.  ...

### A Parameterized Algorithm for Chordal Sandwich [chapter]

Pinar Heggernes, Federico Mancini, Jesper Nederlof, Yngve Villanger
2010 Lecture Notes in Computer Science
In particular, we give an algorithm with running time O(2 k n 5 ) to solve this problem, where k is the size of a minimum vertex cover of the graph (V, F ).  ...  Given an arbitrary graph G = (V, E) and an additional set of admissible edges F , the Chordal Sandwich problem asks whether there exists a chordal graph (V, E ∪ F ) such that F ⊆ F .  ...  By the results of  , we can "search through" all minimal triangulations of a graph if we have a list of all its minimal separators and potential maximal cliques.  ...

### Exact (Exponential) Algorithms for Treewidth and Minimum Fill-In [chapter]

Fedor V. Fomin, Dieter Kratsch, Ioan Todinca
2004 Lecture Notes in Computer Science
Our result is based on a combinatorial proof that the number of minimal separators in a graph is O(n · 1.7087 n ) and that the number of potential maximal cliques is O(n 4 · 1.9601 n ).  ...  We show that for a graph G on n vertices its treewidth and minimum fill-in can be computed roughly in 1.9601 n time.  ...  Fedor Fomin acknowledges support of Norges forskningsråd, project 160778/V30.  ...

### Algorithms parameterized by vertex cover and modular width, through potential maximal cliques [article]

Fedor V. Fomin, Mathieu Liedloff, Pedro Montealegre, Ioan Todinca
2014 arXiv   pre-print
We prove that for any graph, the number of minimal separators is O^*(3^vc) and O^*(1.6181^mw), and the number of potential maximal cliques is O^*(4^vc) and O^*(1.7347^mw), and these objects can be listed  ...  In this paper we give upper bounds on the number of minimal separators and potential maximal cliques of graphs w.r.t. two graph parameters, namely vertex cover (vc) and modular width (mw).  ...  By Proposition 6, this algorithm covers alls cases and thus lists all potential maximal cliques of G i .  ...

### Visualizing plant metabolomic correlation networks using clique-metabolite matrices

F. Kose, W. Weckwerth, T. Linke, O. Fiehn
2001 Bioinformatics
Next, a branch-and-bound algorithm was developed to find all maximal cliques by combining submaximal cliques.  ...  First, a list of linear metabolite correlations is generated that can be regarded as a set of pairs of edges (or as 2-cliques).  ...  All vertices that are members of a clique are numbered from 1 to n. A list of all cliques including n vertices is indicated as C n .  ...

### Treewidth and Minimum Fill-In of Weakly Triangulated Graphs [chapter]

Vincent Bouchitté, Ioan Todinca
1999 Lecture Notes in Computer Science
We introduce the notion of potential maximal clique of a graph and we use it for computing the treewidth and the minimum ll-in of graphs for which the the potential maximal cliques can be listed in polynomial  ...  Finally we show how to compute the potential maximal cliques of weakly triangulated graphs.  ...  One can check in O(n 2 ) time if a set is a potential maximal clique, using theorem 4. So the list of all potential maximal cliques is computable in O(n 2 e).  ...

### Independent Set inP5-Free Graphs in Polynomial Time [chapter]

Daniel Lokshantov, Martin Vatshelle, Yngve Villanger
2013 Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms
In particular the complexity of Independent Set on P 5 -free graphs has received significant attention, and there has been a long list of results showing that the problem becomes polynomial time solvable  ...  on sub-classes of P 5 -free graphs.  ...  The potential maximal cliques of a graph G are defined to be the set of maximal cliques over all minimal triangulations of the graph G.  ...

### Faster Parameterized Algorithms for Minimum Fill-in

Hans L. Bodlaender, Pinar Heggernes, Yngve Villanger
2010 Algorithmica
graph?  ...  To achieve these results, we present a new lemma describing the edges that can safely be added to achieve a chordal completion with the minimum number of edges, regardless of k.  ...  We make a list of potential maximal cliques of G. Again, this list does not need all potential maximal cliques, separators, but it contains all that are needed to make step 5 succeed. 5.  ...

### Faster Parameterized Algorithms for Minimum Fill-In [chapter]

Hans L. Bodlaender, Pinar Heggernes, Yngve Villanger
2008 Lecture Notes in Computer Science
graph?  ...  To achieve these results, we present a new lemma describing the edges that can safely be added to achieve a chordal completion with the minimum number of edges, regardless of k.  ...  We make a list of potential maximal cliques of G. Again, this list does not need all potential maximal cliques, separators, but it contains all that are needed to make step 5 succeed. 5.  ...
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