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47
Complexity classification of some edge modification problems
, 2001
"... In an edge modification problem one has to change the edge set of a given graph as little as possible so as to satisfy a certain property. We prove the NPhardness of a variety of edge modification problems with respect to some wellstudied classes of graphs. These include perfect, chordal, chain, c ..."
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Cited by 41 (2 self)
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In an edge modification problem one has to change the edge set of a given graph as little as possible so as to satisfy a certain property. We prove the NPhardness of a variety of edge modification problems with respect to some wellstudied classes of graphs. These include perfect, chordal, chain, comparability, split and asteroidal triple free. We show that some of these problems become polynomial when the input graph has bounded degree. We also give a general constant factor approximation algorithm for deletion and editing problems on bounded degree graphs with respect to properties that can be characterized by a finite set of forbidden induced subgraphs.
Genomics via Optical Mapping II: Ordered Restriction Maps
 Journal of Computational Biology
, 1996
"... In this paper, we describe our algorithmic approach to constructing ordered restriction maps based on the data created from the images of population of individual DNA molecules (clones) digested by restriction enzymes. The goal is to devise mapmaking algorithms capable of producing highresolution, ..."
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Cited by 33 (19 self)
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In this paper, we describe our algorithmic approach to constructing ordered restriction maps based on the data created from the images of population of individual DNA molecules (clones) digested by restriction enzymes. The goal is to devise mapmaking algorithms capable of producing highresolution, highaccuracy maps rapidly and in a scalable manner. The resulting software is a key component of our optical mapping automation tools and has been used routinely to map cosmid, lambda and BAC clones. The experimental results appear highly promising. 1 Genomics and Optical Mapping Optical mapping [CAH+95, CJI+96, HRL+95, JRH+96, MBC+95, SCH+95, SLH+93, WHS95] is a single molecule methodology for the rapid production of ordered restriction maps from individual DNA molecules. Ordered restriction maps were constructed originally from yeast chromosomes by using fluorescence microscopy to visualize restriction endonuclease cutting events on individual fluorochromestained DNA molecules [SCH+95,...
Pathwidth, Bandwidth and Completion Problems to Proper Interval Graphs with Small Cliques
 SIAM Journal on Computing
, 1996
"... We study two related problems motivated by molecular biology: ffl Given a graph G and a constant k, does there exist a supergraph G of G which is a unit interval graph and has clique size at most k? ffl Given a graph G and a proper kcoloring c of G, does there exist a supergraph We show th ..."
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Cited by 29 (6 self)
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We study two related problems motivated by molecular biology: ffl Given a graph G and a constant k, does there exist a supergraph G of G which is a unit interval graph and has clique size at most k? ffl Given a graph G and a proper kcoloring c of G, does there exist a supergraph We show that those problems are polynomial for fixed k. On the other hand we prove that the first problem is equivalent to deciding if the bandwidth of G is at most k \Gamma 1. Hence, it is NPhard, and W [t]hard for all t. We also show that the second problem is W [1]hard. This implies that for fixed k, both of the problems are unlikely to have an O(n ) algorithm, where ff is a constant independent of k.
A methodological framework for the reconstruction of contiguous regions of ancestral genomes and its application to mammalian genome
 PLoS Comput. Biol
, 1000
"... The reconstruction of ancestral genome architectures and gene orders from homologies between extant species is a longstanding problem, considered by both cytogeneticists and bioinformaticians. A comparison of the two approaches was recently investigated and discussed in a series of papers, sometimes ..."
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Cited by 28 (13 self)
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The reconstruction of ancestral genome architectures and gene orders from homologies between extant species is a longstanding problem, considered by both cytogeneticists and bioinformaticians. A comparison of the two approaches was recently investigated and discussed in a series of papers, sometimes with diverging points of view regarding the performance of these two approaches. We describe a general methodological framework for reconstructing ancestral genome segments from conserved syntenies in extant genomes. We show that this problem, from a computational point of view, is naturally related to physical mapping of chromosomes and benefits from using combinatorial tools developed in this scope. We develop this framework into a new reconstruction method considering conserved gene clusters with similar gene content, mimicking principles used in most cytogenetic studies, although on a different kind of data. We implement and apply it to datasets of mammalian genomes. We perform intensive theoretical and experimental comparisons with other bioinformatics methods for ancestral genome segments reconstruction. We show that the method that we propose is stable and reliable: it gives convergent results using several kinds of data at different levels of resolution, and all predicted ancestral regions are well supported. The results come eventually very close to cytogenetics studies. It suggests that the comparison of methods for ancestral genome reconstruction should include the algorithmic aspects of the methods as well
Visualizing Related Metabolic Pathways in Two and a Half Dimensions
, 2003
"... We propose a method for visualizing a set of related metabolic pathways using 2 2 D graph drawing. Interdependent, twodimensional layouts of each pathway are stacked on top of each other so that biologists get a full picture of subtle and significant di#erences among the pathways. Layouts are ..."
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Cited by 26 (7 self)
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We propose a method for visualizing a set of related metabolic pathways using 2 2 D graph drawing. Interdependent, twodimensional layouts of each pathway are stacked on top of each other so that biologists get a full picture of subtle and significant di#erences among the pathways. Layouts are determined by a global layout of the union of all pathwayrepresenting graphs using a variant of the proven Sugiyama approach for layered graph drawing that allows edges to cross if they appear in di#erent graphs.
Genomics via Optical Mapping III: Contiging Genomic DNA and Variations (Extended Abstract)
, 1997
"... ) Thomas Anantharaman, Bud Mishra and David Schwartz 1 Abstract In this paper, we describe our algorithmic approach to constructing an alignment of (con tiging) a set of optical maps created from the images of individual genomic DNA molecules digested by restriction enzymes. Generally, these DNA ..."
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Cited by 26 (19 self)
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) Thomas Anantharaman, Bud Mishra and David Schwartz 1 Abstract In this paper, we describe our algorithmic approach to constructing an alignment of (con tiging) a set of optical maps created from the images of individual genomic DNA molecules digested by restriction enzymes. Generally, these DNA segments are sized in the range of 14Mb. The problem of assembling clone contig maps is a simpler special case of this contig problem and is handled by our algorithms. The goal is to devise contiging algorithms capable of producing highquality composite maps rapidly and in a scalable manner. The resulting software is a key component of our physical mapping automation tools and has been used routinely to create composite maps of various microorganisms (E. coli, P. falciparum and D. radioduran). The experimental results appear highly promising. 1 Introduction Single molecule approaches provide a new direction for characterizing structural and functional properties of individual DNA molec...
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 25 (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. 1 Introduction and
Towards Constructing Physical Maps by Optical Mapping: An Effective, Simple, Combinatorial Approach (Extended Abstract)
, 1997
"... ) S. Muthukrishnan Laxmi Parida y Abstract We initiate the complexity study of physical mapping with the emerging technology of Optical Mapping (OM) pioneered by the team lead by David Schwartz at the W. M. Keck Laboratory for Biomolecular Imaging, Dept of Chemistry, NYU. In currently popular e ..."
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Cited by 16 (6 self)
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) S. Muthukrishnan Laxmi Parida y Abstract We initiate the complexity study of physical mapping with the emerging technology of Optical Mapping (OM) pioneered by the team lead by David Schwartz at the W. M. Keck Laboratory for Biomolecular Imaging, Dept of Chemistry, NYU. In currently popular electrophoretic approaches, information about the relative ordering of the fragments comprising the DNA molecule is lost, thus leading to difficult computational problems of composing the fragments in to a physical map depicting their relative order. In contrast, the relative ordering of the pieces is readily obtained in OM. However, OM faces serious technological challenges as it has low resolution and is faultprone. We take a combinatorial approach to the problem of constructing physical maps from the erroneous data generated by OM. We identify two abstract problems in this context, namely, the Exclusive Binary FlipCut and Exclusive Weighted FlipCut problems. For both, we present polynom...
Visual understanding of metabolic pathways across organisms using layout in two and a half dimensions
 JOURNAL OF INTEGRATIVE BIOINFORMATICS
, 2004
"... We propose a method for visualizing a set of related metabolic pathways across organisms using 2 1/2 dimensional graph visualization. Interdependent, twodimensional layouts of each pathway are stacked on top of each other so that biologists get a full picture of subtle and significant differences am ..."
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Cited by 13 (8 self)
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We propose a method for visualizing a set of related metabolic pathways across organisms using 2 1/2 dimensional graph visualization. Interdependent, twodimensional layouts of each pathway are stacked on top of each other so that biologists get a full picture of subtle and significant differences among the pathways. The (dis)similarities between pathways are expressed by the Hamming distances of the underlying graphs which are used to compute a stacking order for the pathways. Layouts are determined by a global layout of the union of all pathway graphs using a variant of the proven Sugiyama approach for layered graph drawing. Our variant layout approach allows edges to cross if they appear in different graphs.
Fixedparameter complexity of minimum profile problems
 In Proceedings IWPEC 2006
, 2006
"... The profile of a graph is an integervalued parameter defined via vertex orderings; it is known that the profile of a graph equals the smallest number of edges of an interval supergraph. Since computing the profile of a graph is an NPhard problem, we consider parameterized versions of the problem. ..."
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Cited by 12 (7 self)
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The profile of a graph is an integervalued parameter defined via vertex orderings; it is known that the profile of a graph equals the smallest number of edges of an interval supergraph. Since computing the profile of a graph is an NPhard problem, we consider parameterized versions of the problem. Namely, we study the problem of deciding whether the profile of a connected graph of order n is at most n − 1 + k, considering k as the parameter; this is a parameterization above guaranteed value, since n − 1 is a tight lower bound for the profile. We present two fixedparameter algorithms for this problem. The first algorithm is based on a forbidden subgraph characterization of interval graphs. The second algorithm is based on two simple kernelization rules which allow us to produce a kernel with linear number of vertices and edges. For showing the correctness of the second algorithm we need to establish structural properties of graphs with small profile which are of independent interest. 1