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23
PC trees and circularones arrangements
 Theoretical Computer Science
"... A 01 matrix has the consecutiveones property if its columns can be ordered so that the ones in every row are consecutive. It has the circularones property if its columns can be ordered so that, in every row, either the ones or the zeros are consecutive. PQ trees are used for representing all cons ..."
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Cited by 34 (4 self)
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A 01 matrix has the consecutiveones property if its columns can be ordered so that the ones in every row are consecutive. It has the circularones property if its columns can be ordered so that, in every row, either the ones or the zeros are consecutive. PQ trees are used for representing all consecutiveones orderings of the columns of a matrix that has the consecutiveones property. We give an analogous structure, called a PC tree, for representing all circularones orderings of the columns of a matrix that has the circularones property. No such representation has been given previously. In contrast to PQ trees, PC trees are unrooted. We obtain a much simpler algorithm for computing PQ trees that those that were previously available, by adding a zero column, x, to a matrix, computing the PC tree, and then picking the PC tree up by x to root it. 1
Decidability of String Graphs
 Proceedings of the 33rd Annual Symposium on the Theory of Computing
, 2003
"... We show that string graphs can be recognized in nondeterministic exponential time by giving an exponential upper bound on the number of intersections for a minimal drawing realizing a string graph in the plane. This upper bound confirms a conjecture by Kratochvl and Matousek [KM91] and settles th ..."
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Cited by 24 (5 self)
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We show that string graphs can be recognized in nondeterministic exponential time by giving an exponential upper bound on the number of intersections for a minimal drawing realizing a string graph in the plane. This upper bound confirms a conjecture by Kratochvl and Matousek [KM91] and settles the longstanding open problem of the decidability of string graph recognition (Sinden [Sin66], Graham [Gra76]). Finally we show how to apply the result to solve another old open problem: deciding the existence of Euler diagrams, a fundamental problem of topological inference (Grigni, Papadias, Papadimitriou [GPP95]). The general theory of Euler diagrams turns out to be as hard as secondorder arithmetic.
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 20 (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.
Interval bigraphs and circular arc graphs
 J. Graph Theory
"... Abstract We prove that the complements of interval bigraphs are precisely those circular arc graphs of clique covering number two which admit a representation without two arcs covering the whole circle. We give another characterization of interval bigraphs, in terms of a vertex ordering, that we hop ..."
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Cited by 19 (5 self)
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Abstract We prove that the complements of interval bigraphs are precisely those circular arc graphs of clique covering number two which admit a representation without two arcs covering the whole circle. We give another characterization of interval bigraphs, in terms of a vertex ordering, that we hope may prove helpful in finding a more efficient recognition algorithm than presently known. We use these results to show equality, amongst bipartite graphs, of several classes of structured graphs (proper interval bigraphs, complements of proper circular arc graphs, asteroidaltriplefree graphs, permutation graphs, and cocomparability graphs). Our results verify a conjecture of Lundgren and disprove a conjecture of M"uller. 1 Background A graph H is an interval graph if it is the intersection graph of a family of intervals Iv, v 2 V (H). (Two vertices v; v 0 are adjacent in H if and only if Iv and Iv0 intersect.) If the
Algorithmic Aspects of the ConsecutiveOnes Property
, 2009
"... We survey the consecutiveones property of binary matrices. Herein, a binary matrix has the consecutiveones property (C1P) if there is a permutation of its columns that places the 1s consecutively in every row. We provide an overview over connections to graph theory, characterizations, recognition ..."
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Cited by 7 (1 self)
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We survey the consecutiveones property of binary matrices. Herein, a binary matrix has the consecutiveones property (C1P) if there is a permutation of its columns that places the 1s consecutively in every row. We provide an overview over connections to graph theory, characterizations, recognition algorithms, and applications such as integer linear programming and solving Set Cover.
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 6 (1 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.
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 6 (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.
Pathwidth of circulararc graphs
 Proceedings of WG 2007, Lecture Notes in Computer Science 4769, 2007
"... Abstract. The pathwidth of a graph G is the minimum clique number of H minus one, over all interval supergraphs H of G. Although pathwidth is a wellknown and wellstudied graph parameter, there are extremely few graph classes for which pathwidh is known to be tractable in polynomial time. We give i ..."
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Cited by 5 (0 self)
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Abstract. The pathwidth of a graph G is the minimum clique number of H minus one, over all interval supergraphs H of G. Although pathwidth is a wellknown and wellstudied graph parameter, there are extremely few graph classes for which pathwidh is known to be tractable in polynomial time. We give in this paper an O(n 2)time algorithm computing the pathwidth of circulararc graphs. 1
Linear time recognition of Helly circulararc models and graphs
 manuscript, 2007 (Presented at COCOON’06 and SIAM DM’06 Confs
, 2007
"... Abstract. A circulararc model (C, A) is a circle C together with a collection A of arcs of C. If A satisfies the Helly Property then (C, A) is a Helly circulararc model. A (Helly) circulararc graph is the intersection graph of a (Helly) circulararc model. Circulararc graphs and their subclasses ..."
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Cited by 5 (2 self)
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Abstract. A circulararc model (C, A) is a circle C together with a collection A of arcs of C. If A satisfies the Helly Property then (C, A) is a Helly circulararc model. A (Helly) circulararc graph is the intersection graph of a (Helly) circulararc model. Circulararc graphs and their subclasses have been the object of a great deal of attention, in the literature. Linear time recognition algorithm have been described both for the general class and for some of its subclasses. However, for Helly circulararc graphs, the best recognition algorithm is that by Gavril, whose complexity is O(n 3). In this article, we describe different characterizations for Helly circulararc graphs, including a characterization by forbidden induced subgraphs for the class. The characterizations lead to a linear time recognition algorithm for recognizing graphs of this class. The algorithm also produces certificates for a negative answer, by exhibiting a forbidden subgraph of it, within this same bound.
Solving the canonical representation and star system problems for proper circulararc graphs in logspace. Eprint: http://arxiv.org/abs/1202.4406
, 2012
"... We present a logspace algorithm that constructs a canonical intersection model for a given proper circulararc graph, where canonical means that isomorphic graphs receive identical models. This implies that the recognition and the isomorphism problems for these graphs are solvable in logspace. For t ..."
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Cited by 2 (2 self)
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We present a logspace algorithm that constructs a canonical intersection model for a given proper circulararc graph, where canonical means that isomorphic graphs receive identical models. This implies that the recognition and the isomorphism problems for these graphs are solvable in logspace. For the broader class of concaveround graphs, which still possess (not necessarily proper) circulararc models, we show that a canonical circulararc model can also be constructed in logspace. As a building block for these results, we design a logspace algorithm for computing canonical circulararc models of circulararc hypergraphs; this important class of hypergraphs corresponds to matrices with the circular ones property. Furthermore, we consider the Star System Problem that consists in reconstructing a graph from its closed neighborhood hypergraph. We show that this problem is solvable in logarithmic space for the classes of proper circulararc, concaveround, and coconvex graphs.