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Minimal triangulations of graphs: A survey
 DISCRETE MATHEMATICS
"... Any given graph can be embedded in a chordal graph by adding edges, and the resulting chordal graph is called a triangulation of the input graph. In this paper we study minimal triangulations, which are the result of adding an inclusion minimal set of edges to produce a triangulation. This topic was ..."
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Cited by 35 (3 self)
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Any given graph can be embedded in a chordal graph by adding edges, and the resulting chordal graph is called a triangulation of the input graph. In this paper we study minimal triangulations, which are the result of adding an inclusion minimal set of edges to produce a triangulation. This topic was first studied from the standpoint of sparse matrices and vertex elimination in graphs. Today we know that minimal triangulations are closely related to minimal separators of the input graph. Since the first papers presenting minimal triangulation algorithms appeared in 1976, several characterizations of minimal triangulations have been proved, and a variety of algorithms exist for computing minimal triangulations of both general and restricted graph classes. This survey presents and ties together these results in a unified modern notation, keeping an emphasis on the algorithms.
Rapid protein sidechain packing via tree decomposition
 Research in Computational Molecular Biology, Lecture Notes in Computer Science
, 2005
"... Abstract. This paper proposes a novel tree decomposition based sidechain assignment algorithm, which can obtain the globally optimal solution of the sidechain packing problem very efficiently. Theoretically, the computational complexity of this algorithm is O((N +M)n tw+1 rot) where N is the numbe ..."
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Cited by 26 (1 self)
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Abstract. This paper proposes a novel tree decomposition based sidechain assignment algorithm, which can obtain the globally optimal solution of the sidechain packing problem very efficiently. Theoretically, the computational complexity of this algorithm is O((N +M)n tw+1 rot) where N is the number of residues in the protein, M the number of interacting residue pairs, nrot the average number of rotamers for each residue and tw( = O(N 2 3 log N)) the tree width of the residue interaction graph. Based on this algorithm, we have developed a sidechain prediction program SCATD (Side Chain Assignment via Tree Decomposition). Experimental results show that after the Goldstein DEE is conducted, nrot is around 3.5, tw is only 3 or 4 for most of the test proteins in the SCWRL benchmark and less than 10 for all the test proteins. SCATD runs up to 90 times faster than SCWRL 3.0 on some large proteins in the SCWRL benchmark and achieves an average of five times faster speed on all the test proteins. If only the postDEE stage is taken into consideration, then our treedecomposition based energy minimization algorithm is more than 200 times faster than that in SCWRL 3.0 on some large proteins. SCATD is freely available for academic research upon request. 1
Fast and accurate algorithms for protein sidechain packing
, 2006
"... This article studies the protein sidechain packing problem using the treedecomposition of a protein structure. To obtain fast and accurate protein sidechain packing, protein structures are modeled using a geometric neighborhood graph, which can be easily decomposed into smaller blocks. Therefor ..."
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Cited by 11 (1 self)
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This article studies the protein sidechain packing problem using the treedecomposition of a protein structure. To obtain fast and accurate protein sidechain packing, protein structures are modeled using a geometric neighborhood graph, which can be easily decomposed into smaller blocks. Therefore, the sidechain assignment of the whole protein can be assembled from the assignment of the small blocks. Although we will show that the sidechain packing problem is still NPhard, we can achieve a treedecompositionbased globally optimal algorithm with time complexity of O(Nn tw+1 rot) and several polynomialtime approximation schemes (PTAS), where N is the number of residues contained in the protein, nrot the average number of rotamers for each residue, and tw = O(N 2/3 log N) the treewidth of the protein structure graph. Experimental results indicate that after Goldstein deadend elimination is conducted, nrot is very small and tw is equal to 3 or 4 most of the time. Based on the globally optimal algorithm, we developed a protein sidechain assignment program TreePack, which runs up to 90 times faster than SCWRL 3.0, a widelyused sidechain packing program, on some large test proteins in the SCWRL benchmark database and an average of five times faster on all the test proteins in this database. There are also some realworld
New Upper Bound Heuristics for Treewidth
, 2004
"... In this paper, we introduce and evaluate some heuristics to find an upper bound on the treewidth of a given graph. Each of the heuristics selects the vertices of the graph one by one, building an elimination list. The heuristics differ in the criteria used for selecting vertices. These criteria depe ..."
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Cited by 9 (4 self)
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In this paper, we introduce and evaluate some heuristics to find an upper bound on the treewidth of a given graph. Each of the heuristics selects the vertices of the graph one by one, building an elimination list. The heuristics differ in the criteria used for selecting vertices. These criteria depend on the fillin of a vertex and the related new notion of the fillinexcludingoneneighbor. In several cases, the new heuristics improve the bounds obtained by existing heuristics.
Recognizing and Triangulating Chordal Probe Graphs
 and CRI Technical Report
, 2003
"... A graph G = (V; E) is chordal probe if its vertices can be partitioned into two sets P (probes) and N (nonprobes) where N is a stable set and such that G can be extended to a chordal graph by adding edges between nonprobes. ..."
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Cited by 5 (4 self)
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A graph G = (V; E) is chordal probe if its vertices can be partitioned into two sets P (probes) and N (nonprobes) where N is a stable set and such that G can be extended to a chordal graph by adding edges between nonprobes.
Cyclebicolorable graphs and triangulating chordal probe graphs
, 2003
"... A graph G = (V; E) is chordal probe if its vertices can be partitioned into two sets P (probes) and N (nonprobes) where N is a stable set and such that G can be extended to a chordal graph by adding edges between nonprobes. We give several characterizations of chordal probe graphs, first in the ca ..."
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Cited by 4 (2 self)
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A graph G = (V; E) is chordal probe if its vertices can be partitioned into two sets P (probes) and N (nonprobes) where N is a stable set and such that G can be extended to a chordal graph by adding edges between nonprobes. We give several characterizations of chordal probe graphs, first in the case of a fixed given partition of the vertices into probes and nonprobes, and second in the more general case where no partition is given. In both of these cases, our results are obtained by introducing new classes, namely, Ntriangulatable graphs and cyclebicolorable graphs. We give polynomial time recognition algorithms for each class. Ntriangulatable graphs have properties similar to chordal graphs, and we characterize them using graph separators and using a vertex elimination ordering. Cyclebicolorable graphs are shown to be perfect, and any cyclebicoloring of a graph renders it Ntriangulatable. The corresponding recognition complexity for chordal probe graphs, given a partition of the vertices into probes and nonprobes, is O(jP jjEj), thus also providing a interesting tractible subcase of the chordal graph sandwich problem. If no partition is given in advance, the complexity of our recognition algorithm is O(jV j 2
Graph Extremities and Minimal Separation
 Proceedings of JIM 2003
, 2003
"... Many problems related to knowledge discovery can be modelized by an undirected graph, which in turn has a strong structure associated with its minimal separators. On several wellknown graph classes which are used in applications, the graph extremities and the related elimination orderings are an ef ..."
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Cited by 2 (1 self)
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Many problems related to knowledge discovery can be modelized by an undirected graph, which in turn has a strong structure associated with its minimal separators. On several wellknown graph classes which are used in applications, the graph extremities and the related elimination orderings are an efficient algorithmic tool. Often, these extremities turn out to be related to the minimal separators of the graph. We examine this relationship on chordal graphs, weakly chordal graphs, chordal probe graphs, unbreakable graphs and cobipartite graphs, as well as on arbitrary graphs. We show that there are two kinds of extremities: vertices whose neighborhood is a minimal separator (we call these strong extremities) and vertices whose neighborhood looks like a minimal separator (we call these weak extremities).
Reciprocal expressions and the concept of reciprocity. Linguistics and Philosophy
 in &quot;GraphTheoretic Concepts in Computer Science : 31st International Workshop (WG 2005), Metz (France)&quot;, Lecture Notes in Computer Science
, 1998
"... Abstract In this paper, we present a new algorithm for computing the chromatic polynomial of a general graph G. Our method is based on the addition of edges and contraction of nonedges of G, the base case of the recursion being chordal graphs. The set of edges to be considered is taken from a trian ..."
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Abstract In this paper, we present a new algorithm for computing the chromatic polynomial of a general graph G. Our method is based on the addition of edges and contraction of nonedges of G, the base case of the recursion being chordal graphs. The set of edges to be considered is taken from a triangulation of G. To achieve our goal, we use the properties of triangulations and cliquetrees with respect to the previous operations, and guide our algorithm to efficiently divide the original problem. Furthermore, we give some lower bounds of the general complexity of our method, and provide experimental results for several families of graphs. Finally, we exhibit an original measure of a triangulation of a graph.
Characterizing and computing minimal cograph completions
"... A cograph completion of an arbitrary graph G is a cograph supergraph of G on the same vertex set. Such a completion is called minimal if the set of edges added to G is inclusion minimal. In this paper we present two results on minimal cograph completions. The first is a a characterization that allow ..."
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A cograph completion of an arbitrary graph G is a cograph supergraph of G on the same vertex set. Such a completion is called minimal if the set of edges added to G is inclusion minimal. In this paper we present two results on minimal cograph completions. The first is a a characterization that allows us to check in linear time whether a given cograph completion is minimal. The second result is a vertex incremental algorithm to compute a minimal cograph completion H of an arbitrary input graph G in O(V (H)  + E(H)) time.