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Listing all potential maximal cliques of a graph
2002
Theoretical Computer Science
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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. ...
doi:10.1016/s0304-3975(01)00007-x
fatcat:isrmixwjw5hwbjyzfea2qsmg5y
Listing All Potential Maximal Cliques of a Graph
[chapter]
2000
Lecture Notes in Computer Science
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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. ...
doi:10.1007/3-540-46541-3_42
fatcat:zraegbzjyzfkxdt4xe26icjtla
Treewidth Computation and Extremal Combinatorics
[chapter]
2008
Lecture Notes in Computer Science
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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 ...
doi:10.1007/978-3-540-70575-8_18
fatcat:rufxi7qxhbdkta6wdbk77nd4di
Treewidth computation and extremal combinatorics
2012
Combinatorica
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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. ...
doi:10.1007/s00493-012-2536-z
fatcat:mzon5tambrdale4mibhrgluw4i
Improved Exponential-Time Algorithms for Treewidth and Minimum Fill-In
[chapter]
2006
Lecture Notes in Computer Science
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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. ...
doi:10.1007/11682462_73
fatcat:4z4rqvaenbejpd5i32aiej5u6m
Treewidth computation and extremal combinatorics
[article]
2008
arXiv
pre-print
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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. ...
arXiv:0803.1321v2
fatcat:vt5vjd4trzhipmzfqnoy4aikwm
Exact Algorithms for Treewidth and Minimum Fill-In
2008
SIAM journal on computing (Print)
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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. ...
doi:10.1137/050643350
fatcat:vp2ffushcfhcnprue4fftvhalu
A Parameterized Algorithm for Chordal Sandwich
[chapter]
2010
Lecture Notes in Computer Science
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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 [13] , we can "search through" all minimal triangulations of a graph if we have a list of all its minimal separators and potential maximal cliques. ...
doi:10.1007/978-3-642-13073-1_12
fatcat:3v5u75m63bfnbnclrvpjvh2leu
Exact (Exponential) Algorithms for Treewidth and Minimum Fill-In
[chapter]
2004
Lecture Notes in Computer Science
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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. ...
doi:10.1007/978-3-540-27836-8_49
fatcat:yz7xvqzjyzhntc7vvlcey73mi4
Algorithms parameterized by vertex cover and modular width, through potential maximal cliques
[article]
2014
arXiv
pre-print
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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 . ...
arXiv:1404.3882v1
fatcat:vm2q5erf25cmhi2ipjnfjvhj2u
Visualizing plant metabolomic correlation networks using clique-metabolite matrices
2001
Bioinformatics
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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 . ...
doi:10.1093/bioinformatics/17.12.1198
pmid:11751228
fatcat:k4wsx2pyfnc7rjj55owezxohhu
Treewidth and Minimum Fill-In of Weakly Triangulated Graphs
[chapter]
1999
Lecture Notes in Computer Science
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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). ...
doi:10.1007/3-540-49116-3_18
fatcat:fnqsieoml5ftjbaqvg7azvs3xu
Independent Set inP5-Free Graphs in Polynomial Time
[chapter]
2013
Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms
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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. ...
doi:10.1137/1.9781611973402.43
dblp:conf/soda/LokshantovVV14
fatcat:jtvkt62z6jeurbs7nlylyh376q
Faster Parameterized Algorithms for Minimum Fill-in
2010
Algorithmica
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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. ...
doi:10.1007/s00453-010-9421-1
fatcat:ejekaaqm3ngojn34p4nvq4rjzy
Faster Parameterized Algorithms for Minimum Fill-In
[chapter]
2008
Lecture Notes in Computer Science
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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. ...
doi:10.1007/978-3-540-92182-0_27
fatcat:j52wbzpw25ftjj4il5dufim2oa
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