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"... A rigid interval graph is an interval graph which has only one clique tree. In 2009, Panda and Das show that all connected unit interval graphs are rigid interval graphs. Generalizing the two classic graph search algorithms, Lexicographic BreadthFirst Search (LBFS) and Maximum Cardinality Search (M ..."
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Cited by 56 (4 self)
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A rigid interval graph is an interval graph which has only one clique tree. In 2009, Panda and Das show that all connected unit interval graphs are rigid interval graphs. Generalizing the two classic graph search algorithms, Lexicographic BreadthFirst Search (LBFS) and Maximum Cardinality Search (MCS), Corneil and Krueger propose in 2008 the socalled Maximal Neighborhood Search (MNS) and show that one sweep of MNS is enough to recognize chordal graphs. We develop the MNS properties of rigid interval graphs and characterize this graph class in several different ways. This allows us obtain several linear time multisweep MNS algorithms for recognizing rigid interval graphs and unit interval graphs, generalizing a corresponding 3sweep LBFS algorithm for unit interval graph recognition designed by Corneil in 2004. For unit interval graphs, we even present a new linear time 2sweep MNS certifying recognition algorithm. Submitted:
LinearTime Recognition of CircularArc Graphs
 Algorithmica
, 2003
"... A graph G is a circulararc graph if it is the intersection graph of a set of arcs on a circle. That is, there is one arc for each vertex of G, and two vertices are adjacent in G if and only if the corresponding arcs intersect. We give a lineartime algorithm for recognizing this class of graphs. W ..."
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Cited by 36 (7 self)
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A graph G is a circulararc graph if it is the intersection graph of a set of arcs on a circle. That is, there is one arc for each vertex of G, and two vertices are adjacent in G if and only if the corresponding arcs intersect. We give a lineartime algorithm for recognizing this class of graphs. When G is a member of the class, the algorithm gives a certificate in the form of a set of arcs that realize it.
Certifying algorithms for recognizing interval graphs and permutation graphs
 SIAM J. COMPUT
, 2006
"... A certifying algorithm for a problem is an algorithm that provides a certificate with each answer that it produces. The certificate is a piece of evidence that proves that the answer has not been compromised by a bug in the implementation. We give lineartime certifying algorithms for recognition o ..."
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Cited by 32 (7 self)
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A certifying algorithm for a problem is an algorithm that provides a certificate with each answer that it produces. The certificate is a piece of evidence that proves that the answer has not been compromised by a bug in the implementation. We give lineartime certifying algorithms for recognition of interval graphs and permutation graphs, and for a few other related problems. Previous algorithms fail to provide supporting evidence when they claim that the input graph is not a member of the class. We show that our certificates of nonmembership can be authenticated in O(V) time.
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
PARTITION REFINEMENT TECHNIQUES: AN INTERESTING ALGORITHMIC TOOL KIT
 INTERNATIONAL JOURNAL OF FOUNDATIONS OF COMPUTER SCIENCE
, 1999
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A Note on the Consecutive Ones Submatrix Problem
, 2002
"... A binary matrix has the Consecutive Ones Property (C1P) for columns if there exists a permutation of its rows that leaves the 1's consecutive in every column. The problem of Consecutive Ones Property for a matrix is a special variant of Consecutive Ones Submatrix problem in which a positive integer ..."
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Cited by 17 (0 self)
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A binary matrix has the Consecutive Ones Property (C1P) for columns if there exists a permutation of its rows that leaves the 1's consecutive in every column. The problem of Consecutive Ones Property for a matrix is a special variant of Consecutive Ones Submatrix problem in which a positive integer K is given and we want to know if there exists a submatrix B of A consisting of K columns of A with C1P property. This paper presents an error in the proof of NPcompleteness for this problem in the reference cited in text by Garey and Johnson [Computers and Intractability, A Guide to the Theory of NPCompleteness, 1979]. 2001 Elsevier Science B.V. All rights reserved.
Certifying LexBFS recognition algorithms for proper inteval graphs and proper interval bigraphs
 SIAM J. Discrete Math
"... Recently, D. Corneil found a simple 3sweep lexicographic breadth first search (LexBFS) algorithm for the recognition of proper interval graphs. We point out how to modify Corneil’s algorithm to make it a certifying algorithm, and then describe a similar certifying 3sweep LexBFS algorithm for the r ..."
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Cited by 17 (3 self)
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Recently, D. Corneil found a simple 3sweep lexicographic breadth first search (LexBFS) algorithm for the recognition of proper interval graphs. We point out how to modify Corneil’s algorithm to make it a certifying algorithm, and then describe a similar certifying 3sweep LexBFS algorithm for the recognition of proper interval bigraphs. It follows from an earlier paper that the class of proper interval bigraphs is equal to the better known class of bipartite permutation graphs, and so we have a certifying algorithm for that class as well. All our algorithms run in time O(m + n), including the certification phase. The certificates of representability (the intervals) can be authenticated in time O(m + n), the certificates of nonrepresentability (the forbidden subgraphs) can be authenticated in time O(n). 1
A simple lineartime modular decomposition algorithm for graphs, using order extension
, 2004
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Graph Decompositions and Factorizing Permutations
 Discrete Mathematics and Theoretical Computer Science
, 1997
"... A factorizing permutation of a given undirected graph is simply a permutation of the vertices in which all decomposition sets appear to be factors. Such a concept seems to play a central role in recent papers dealing with graph decomposition. It is applied here for modular decomposition and we propo ..."
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Cited by 11 (6 self)
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A factorizing permutation of a given undirected graph is simply a permutation of the vertices in which all decomposition sets appear to be factors. Such a concept seems to play a central role in recent papers dealing with graph decomposition. It is applied here for modular decomposition and we propose a linear algorithm that computes the whole decomposition tree when a factorizing permutation is provided. This algorithm can be seen as a common generalization of Ma and Hsu [9, 8] for modular decomposition of chordal graphs and Habib, Huchard and Spinrad [7] for inheritance graphs decomposition. It also suggests many new decomposition algorithms for various notions of graph decompositions.
The consecutive ones submatrix problem for sparse matrices
 Algorithmica
, 2004
"... A 01 matrix has the Consecutive Ones Property (C1P) if there is a permutation of its columns that leaves the 1’s consecutive in each row. The Consecutive Ones Submatrix (C1S) problem is, given a 01 matrix A, to find the largest number of columns of A that form a submatrix with the C1P property. Su ..."
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Cited by 11 (0 self)
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A 01 matrix has the Consecutive Ones Property (C1P) if there is a permutation of its columns that leaves the 1’s consecutive in each row. The Consecutive Ones Submatrix (C1S) problem is, given a 01 matrix A, to find the largest number of columns of A that form a submatrix with the C1P property. Such a problem finds application in physical mapping with hybridization data in genome sequencing. Let (a, b)matrices be the 01 matrices in which there are at most a 1’s in each column and at most b 1’s in each row. This paper proves that the C1S problem remains NPhard for i) (2, 3)matrices and ii) (3, 2)matrices. This solves an open problem posed in a recent paper of Hajiaghayi and Ganjali [1]. We further prove that the C1S problem is polynomialtime 0.8approximatable for (2, 3)matrices in which no two columns are identical and 0.5approximatable for (2, ∞)matrices in general. we also show that the C1S problem is polynomialtime 0.5approximatable for (3, 2)matrices. However, there exists an ɛ> 0 such that approximating the C1S problem for (∞, 2)matrices within a factor of n ɛ (where n is the number of columns of the input matrix) is NPhard. Keywords: NPhardness, approximation algorithm, consecutive ones property, consecutive ones submatrix, caterpillar spanning tree 1