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22
Cliquedetection Models in Computational Biochemistry and Genomics
 European Journal of Operational Research
, 2005
"... Many important problems arising in computational biochemistry and genomics have been formulated in terms of underlying combinatorial optimization models. In particular, a number have been formulated as cliquedetection models. The proposed article includes an introduction to the underlying biochemis ..."
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Cited by 31 (3 self)
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Many important problems arising in computational biochemistry and genomics have been formulated in terms of underlying combinatorial optimization models. In particular, a number have been formulated as cliquedetection models. The proposed article includes an introduction to the underlying biochemistry and genomic aspects of the problems as well as to the graphtheoretic aspects of the solution approaches. Each subsequent section describes a particular type of problem, gives an example to show how the graph model can be derived, summarizes recent progress, and discusses challenges associated with solving the associated graphtheoretic models. Clique detection models include prescribing (a) a maximal clique, (b) a maximum clique, (c) a maximum weighted clique, or (d) all maximal cliques in a graph. The particular types of biochemistry and genomics problems that can be represented by a clique detection model include integration of genome mapping data, nonoverlapping local alignments, matching and comparing molecular structures, and protein docking.
Construction of Probe Interval Models
"... An interval graph for a set of intervals on a line consists of one vertex for each interval, and an edge for each pair of intersecting intervals. A probe interval graph is obtained from an interval graph by designating a subset P of vertices as probes, and removing the edges between pairs of vertice ..."
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Cited by 25 (5 self)
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An interval graph for a set of intervals on a line consists of one vertex for each interval, and an edge for each pair of intersecting intervals. A probe interval graph is obtained from an interval graph by designating a subset P of vertices as probes, and removing the edges between pairs of vertices in the remaining set N of nonprobes. We examine the problem of finding and representing possible layouts of the intervals, given a probe interval graph. We obtain an O(n + m log n) bound, where n is the number of vertices and m is the number of edges. The problem is motivated by an application to molecular biology.
Tree Spanners for Bipartite Graphs and Probe Interval Graphs
, 2003
"... A tree tspanner T in a graph G is a spanning tree of G such that the distance between every pair of vertices in T is at most t times their distance in G. The tree tspanner problem asks whether a graph admits a tree tspanner, given t. We first substantially strengthen the known results for bip ..."
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Cited by 8 (3 self)
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A tree tspanner T in a graph G is a spanning tree of G such that the distance between every pair of vertices in T is at most t times their distance in G. The tree tspanner problem asks whether a graph admits a tree tspanner, given t. We first substantially strengthen the known results for bipartite graphs. We prove that the tree tspanner problem is NPcomplete even for chordal bipartite graphs for t 5, and every bipartite ATEfree graph has a tree 3spanner, which can be found in linear time. The best known before results were NPcompleteness for general bipartite graphs, and that every convex graph has a tree 3spanner. We next focus on the tree tspanner problem for probe interval graphs and related graph classes. The graph classes were introduced to deal with the physical mapping of DNA. From a graph theoretical point of view, the classes are natural generalizations of interval graphs. We show that these classes are tree 7spanner admissible, and a tree 7spanner can be constructed in O(m log n) time.
Algebraic Operations on PQ Trees and Modular Decomposition Trees
, 2005
"... Partitive set families are families of sets that can be quite large, but have a compact, recursive representation in the form of a tree. This tree is a common generalization of PQ trees, the modular decomposition of graphs, certain decompositions of boolean functions, and decompositions that arise o ..."
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Cited by 7 (0 self)
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Partitive set families are families of sets that can be quite large, but have a compact, recursive representation in the form of a tree. This tree is a common generalization of PQ trees, the modular decomposition of graphs, certain decompositions of boolean functions, and decompositions that arise on a variety of other combinatorial structures. We describe natural operators on partitive set families, give algebraic identities for manipulating them, and describe efficient algorithms for evaluating them. We use these results to obtain new time bounds for finding the common intervals of a set of permutations, finding the modular decomposition of an edgecolored graphs (also known as a twostructure), finding the PQ tree of a matrix when a consecutiveones arrangement is given, and finding the modular decomposition of a permutation graph when its permutation realizer is given.
Discovering Temporal Relations in Molecular Pathways Using ProteinProtein Interactions
 Proceedings of the 8th Annual International Conference on Computational Molecular Biology (RECOMB), 2004
, 2004
"... The availability of largescale proteinprotein interaction data provides us with many opportunities to study molecular pathways involving proteins. In this paper we propose to mine temporal relations in molecular pathways by proteinprotein interaction data. In particular, we model the assembly path ..."
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Cited by 5 (0 self)
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The availability of largescale proteinprotein interaction data provides us with many opportunities to study molecular pathways involving proteins. In this paper we propose to mine temporal relations in molecular pathways by proteinprotein interaction data. In particular, we model the assembly pathways of protein complexes with interval graphs and determine the temporal order of joining the pathway for proteins by ordering the vertices in the interaction graph. We develop a tool called Xronos to perform such a computation. We then apply Xronos to the ribosome assembly pathway and present validation results for the obtained ordering. The results are promising and show the potential usage for Xronos in the study of molecular pathways.
2tree probe interval graphs have a large obstruction set
, 2004
"... Probe interval graphs are used as a generalization of interval graphs in physical mapping of DNA. is a probe interval graph (PIG) with respect to a partition ¥���¨��� � of ¦ if vertices of ¢ correspond to intervals on a real line and two vertices are adjacent if and only if their corresponding inter ..."
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Cited by 5 (0 self)
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Probe interval graphs are used as a generalization of interval graphs in physical mapping of DNA. is a probe interval graph (PIG) with respect to a partition ¥���¨��� � of ¦ if vertices of ¢ correspond to intervals on a real line and two vertices are adjacent if and only if their corresponding intervals intersect and at least one of them is in �; vertices belonging to � are called probes and vertices belonging to � are called nonprobes. One common approach to studying the structure of a new family of graphs is to determine if there is a concise family of forbidden induced subgraphs. It has been shown that there are two forbidden induced subgraphs that characterize tree PIGs. In this paper we show that having a concise forbidden induced subgraph characterization does not extend to �tree PIGs; in particular we show that there are at least sixtytwo minimal forbidden induced subgraphs for �tree PIGs.
LinearTime Recognition of Probe Interval Graphs
"... Abstract. The interval graph for a set of intervals on a line consists of one vertex for each interval, and an edge for each intersecting pair of intervals. A probe interval graph is a variant that is motivated by an application to genomics, where the intervals are partitioned into two sets: probes ..."
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Cited by 3 (1 self)
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Abstract. The interval graph for a set of intervals on a line consists of one vertex for each interval, and an edge for each intersecting pair of intervals. A probe interval graph is a variant that is motivated by an application to genomics, where the intervals are partitioned into two sets: probes and nonprobes. The graph has an edge between two vertices if they intersect and at least one of them is a probe. We give a lineartime algorithm for determining whether a given graph and partition of vertices into probes and nonprobes is a probe interval graph. If it is, we give a layout of intervals that proves that it is. In contrast to previous algorithms for the problem, our algorithm can determine whether the layout is uniquely constrained. This is important for the biological application, where one seeks the true layout of the intervals in a genome. As part of the algorithm we solve the consecutiveones probe matrix problem. 1