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20
Four Strikes against Physical Mapping of DNA
 JOURNAL OF COMPUTATIONAL BIOLOGY
, 1993
"... Physical Mapping is a central problem in molecular biology ... and the human genome project. The problem is to reconstruct the relative position of fragments of DNA along the genome from information on their pairwise overlaps. We show that four simplified models of the problem lead to NPcomplete ..."
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Cited by 55 (8 self)
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Physical Mapping is a central problem in molecular biology ... and the human genome project. The problem is to reconstruct the relative position of fragments of DNA along the genome from information on their pairwise overlaps. We show that four simplified models of the problem lead to NPcomplete decision problems: Colored unit interval graph completion, the maximum interval (or unit interval) subgraph, the pathwidth of a bipartite graph, and the kconsecutive ones problem for k >= 2. These models have been chosen to reflect various features typical in biological data, including false negative and positive errors, small width of the map and chimericism.
Tractability of Parameterized Completion Problems on Chordal, Strongly Chordal and Proper Interval Graphs
, 1994
"... We study the parameterized complexity of three NPhard graph completion problems. The MINIMUM FILLIN problem is to decide if a graph can be triangulated by adding at most k edges. We develop O(c m) and O(k mn + f(k)) algorithms for this problem on a graph with n vertices and m edges. Here f(k ..."
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Cited by 40 (5 self)
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We study the parameterized complexity of three NPhard graph completion problems. The MINIMUM FILLIN problem is to decide if a graph can be triangulated by adding at most k edges. We develop O(c m) and O(k mn + f(k)) algorithms for this problem on a graph with n vertices and m edges. Here f(k) is exponential in k and the constants hidden by the bigO notation are small and do not depend on k. In particular, this implies that the problem is fixedparameter tractable (FPT). The PROPER
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.
Simple Linear Time Recognition of Unit Interval Graphs
, 1998
"... We present a linear time algorithm for unit interval graph recognition. The algorithm is simple and based on BreadthFirst Search. It is also direct  it does not first recognize the graph as an interval graph. Given a graph G, the algorithm produces an ordering of the vertices of the graph whe ..."
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Cited by 29 (1 self)
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We present a linear time algorithm for unit interval graph recognition. The algorithm is simple and based on BreadthFirst Search. It is also direct  it does not first recognize the graph as an interval graph. Given a graph G, the algorithm produces an ordering of the vertices of the graph whenever G is a unit interval graph. This order corresponds to the order of the intervals of some unit interval model for G when arranged according to the increasing order of their left end coordinates. BreadthFirst Search can also be used to construct a unit interval model for a unit interval graph on n vertices; in this model each endpoint is rational, with denominator n. Keywords: graph algorithms, interval graphs, BreadthFirst Search, unit interval graphs, proper interval graphs, design of algorithms. 1 Introduction A graph G is an interval graph if its vertices can be put in a one to one correspondence with a family of intervals I on the real line such that two vertices in G are a...
Generalizations of tournaments: A survey
 J. Graph Theory
, 1998
"... We survey results concerning various generalizations of tournaments. The reader will see that tournaments are by no means the only class of directed graphs with a very rich structure. We describe, among numerous other topics mostly related to paths and cycles, results on hamiltonian paths and cycles ..."
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Cited by 25 (11 self)
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We survey results concerning various generalizations of tournaments. The reader will see that tournaments are by no means the only class of directed graphs with a very rich structure. We describe, among numerous other topics mostly related to paths and cycles, results on hamiltonian paths and cycles. The reader will see that although these problems are polynomially solvable for all of the classes described, they can be highly nontrivial, even for these ”tournamentlike ” digraphs. 1
A simple 3sweep LBFS algorithm for the recognition of unit interval graphs
 DISCRETE APPLIED MATHEMATICS
, 2003
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Clique graphs of Helly circulararc graphs
 Ars Combinatoria
"... Abstract: Clique graphs of several classes of graphs have been already characterized. Trees, interval graphs, chordal graphs, block graphs, cliqueHelly graphs are some of them. However, no characterization of clique graphs of circulararc graphs and some of their subclasses is known. In this paper, ..."
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Cited by 8 (3 self)
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Abstract: Clique graphs of several classes of graphs have been already characterized. Trees, interval graphs, chordal graphs, block graphs, cliqueHelly graphs are some of them. However, no characterization of clique graphs of circulararc graphs and some of their subclasses is known. In this paper, we present a characterization theorem of clique graphs of Helly circulararc graphs and prove that this subclass of circulararc graphs is contained in the intersection between proper circulararc graphs, cliqueHelly circulararc graphs and Helly circulararc graphs. Furthermore, we prove properties about the 2nd iterated clique graph of this family of graphs. Keywords: Circulararc graphs, clique graphs, Helly circulararc graphs, intersection graphs. 1
Proper Helly CircularArc Graphs
"... A circulararc model M = (C, A) is a circle C together with a collection A of arcs of C. If no arc is contained in any other then M is a proper circulararc model, if every arc has the same length then M is a unit circulararc model and if A satisfies the Helly Property then M is a Helly circularar ..."
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Cited by 7 (4 self)
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A circulararc model M = (C, A) is a circle C together with a collection A of arcs of C. If no arc is contained in any other then M is a proper circulararc model, if every arc has the same length then M is a unit circulararc model and if A satisfies the Helly Property then M is a Helly circulararc model. A (proper) (unit) (Helly) circulararc graph is the intersection graph of the arcs of a (proper) (Helly) circulararc model. Circulararc graphs and their subclasses have been the object of a great deal of attention in the literature. Linear time recognition algorithms have been described both for the general class and for some of its subclasses. In this article we study the circulararc graphs which admit a model which is simultaneously proper and Helly. We describe characterizations for this class, including one by forbidden induced subgraphs. These characterizations lead to linear time certifying algorithms for recognizing such graphs. Furthermore, we extend the results to graphs which admit a model which is simultaneously unit and Helly, also leading to characterizations and a linear time certifying algorithm. Key words: algorithms, forbidden subgraphs, Helly circulararc graphs, proper circulararc graphs, unit circulararc graphs.
Satisfiability Problems on Intervals and Unit Intervals
 Theoretical Computer Science
, 1997
"... For an interval graph with some additional order constraints between pairs of nonintersecting intervals, we give a linear time algorithm to determine if there exists a realization which respects the order constraints. Previous algorithms for this problem (known also as seriation with side constrain ..."
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Cited by 5 (1 self)
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For an interval graph with some additional order constraints between pairs of nonintersecting intervals, we give a linear time algorithm to determine if there exists a realization which respects the order constraints. Previous algorithms for this problem (known also as seriation with side constraints) required quadratic time. This problem contains as subproblems interval graph and interval order recognition. On the other hand, it is a special case of the interval satisfiability problem, which is concerned with the realizability of a set of intervals along a line, subject to precedence and intersection constraints. We study such problems for all possible restrictions on the types of constraints, when all intervals must have the same length. We give efficient algorithms for several restrictions of the problem, and show the NPcompleteness of another restriction. 1 Introduction Two intervals x; y on the real line may either intersect or one of them is completely to the left of the othe...