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24
Complexity and Algorithms for Reasoning About Time: A GraphTheoretic Approach
, 1992
"... Temporal events are regarded here as intervals on a time line. This paper deals with problems in reasoning about such intervals when the precise topological relationship between them is unknown or only partially specified. This work unifies notions of interval algebras in artificial intelligence ..."
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Cited by 86 (11 self)
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Temporal events are regarded here as intervals on a time line. This paper deals with problems in reasoning about such intervals when the precise topological relationship between them is unknown or only partially specified. This work unifies notions of interval algebras in artificial intelligence with those of interval orders and interval graphs in combinatorics. The satisfiability, minimal labeling, all solutions and all realizations problems are considered for temporal (interval) data. Several versions are investigated by restricting the possible interval relationships yielding different complexity results. We show that even when the temporal data comprises of subsets of relations based on intersection and precedence only, the satisfiability question is NPcomplete. On the positive side, we give efficient algorithms for several restrictions of the problem. In the process, the interval graph sandwich problem is introduced, and is shown to be NPcomplete. This problem is als...
Parallel Algorithms for Hierarchical Clustering and Applications to Split Decomposition and Parity Graph Recognition
 JOURNAL OF ALGORITHMS
, 1998
"... We present efficient (parallel) algorithms for two hierarchical clustering heuristics. We point out that these heuristics can also be applied to solve some algorithmic problems in graphs. This includes split decomposition. We show that efficient parallel split decomposition induces an efficient para ..."
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Cited by 24 (1 self)
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We present efficient (parallel) algorithms for two hierarchical clustering heuristics. We point out that these heuristics can also be applied to solve some algorithmic problems in graphs. This includes split decomposition. We show that efficient parallel split decomposition induces an efficient parallel parity graph recognition algorithm. This is a consequence of the result of [7] that parity graphs are exactly those graphs that can be split decomposed into cliques and bipartite graphs.
Temporal Constraints: A Survey
, 1998
"... . Temporal Constraint Satisfaction is an information technology useful for representing and answering queries about the times of events and the temporal relations between them. Information is represented as a Constraint Satisfaction Problem (CSP) where variables denote event times and constraints re ..."
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Cited by 20 (1 self)
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. Temporal Constraint Satisfaction is an information technology useful for representing and answering queries about the times of events and the temporal relations between them. Information is represented as a Constraint Satisfaction Problem (CSP) where variables denote event times and constraints represent the possible temporal relations between them. The main tasks are two: (i) deciding consistency, and (ii) answering queries about scenarios that satisfy all constraints. This paper overviews results on several classes of Temporal CSPs: qualitative interval, qualitative point, metric point, and some of their combinations. Research has progressed along three lines: (i) identifying tractable subclasses, (ii) developing exact search algorithms, and (iii) developing polynomialtime approximation algorithms. Most available techniques are based on two principles: (i) enforcing local consistency (e.g. pathconsistency), and (ii) enhancing naive backtracking search. Keywords: Temporal Constra...
Representing Graphs by Disks and Balls (a survey of recognitioncomplexity results)
"... . Practical applications, like radio frequency assignments, led to the denition of disk intersection graphs in the plane, called shortly disk graphs. If the disks in the representation are not allowed to overlap, we speak about disk contact graphs (coin graphs). In this paper we survey recogniti ..."
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Cited by 12 (1 self)
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. Practical applications, like radio frequency assignments, led to the denition of disk intersection graphs in the plane, called shortly disk graphs. If the disks in the representation are not allowed to overlap, we speak about disk contact graphs (coin graphs). In this paper we survey recognitioncomplexity results for disk intersection and contact graphs in the plane. In particular, we refer a classical result by Koebe about disk contact representations, and works of Breu and Kirkpatrick about boundedratio disk representations. We prove that the recognition of diskintersection graphs (in the unbounded ratio case) is NPhard. This result is proved in a more general setting of noncrossing arcconnected sets. We also show some partial results concerning recognition of ball intersection and contact graphs in higher dimensions. In particular, we prove that the recognition of unitball contact graphs is NPhard in dimensions 3; 4, and 8 (24). 1 Introduction 1.1 Intersect...
Circle graphs and Monadic Secondorder logic
, 2005
"... A circle graph is the intersection graph of a set of chords of a circle. If a circle graph is prime for the split (or join) decomposition defined by Cunnigham, it has a unique representation as a set of intersecting chords, and we prove that this representation can be defined by monadic secondorder ..."
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Cited by 9 (5 self)
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A circle graph is the intersection graph of a set of chords of a circle. If a circle graph is prime for the split (or join) decomposition defined by Cunnigham, it has a unique representation as a set of intersecting chords, and we prove that this representation can be defined by monadic secondorder formulas. By using the (canonical) split decomposition of a circle graph, one can define in monadic secondorder logic all its chord representations formalized as words with two occurrences of each letter. This construction uses the general result that the split decomposition of a graph can be constructed in monadic secondorder logic. As a consequence we prove that a set of circle graphs has bounded cliquewidth if and only if all their chord diagrams have bounded treewidth. We also prove that the order of first occurrences of the letters in a double occurrence word w representing a given connected circle graph determines this word w in a unique way.
Contact Graphs of Curves (Extended Abstract)
"... . Contact graphs are a special kind of intersection graphs of geometrical objects in which we do not allow the objects to cross but only to touch each other. Contact graphs of simple curves (and line segments as a special case) in the plane are considered. Several classes of contact graphs are intro ..."
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Cited by 8 (5 self)
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. Contact graphs are a special kind of intersection graphs of geometrical objects in which we do not allow the objects to cross but only to touch each other. Contact graphs of simple curves (and line segments as a special case) in the plane are considered. Several classes of contact graphs are introduced and their properties and inclusions between them are studied. Also the relation between planar and contact graphs is mentioned. Finally, it is proved that the recognition of contact graphs of curves (line segments) is NPcomplete (NPhard) even for planar graphs. 1 Introduction The intersection graphs of geometrical objects have been extensively studied for their many practical applications. Formally the intersection graph of a set family M is defined as a graph G with the vertex set V (G) = M and the edge set E(G) = \Phi fA; Bg ` M j A 6= B; A " B 6= ; \Psi . Probably the first type studied were interval graphs (intersection graphs of intervals on a line), owing to their applic...
The monadic secondorder logic of graphs XVI: Canonical graph decompositions
 Logical Methods in Computer Science
, 2006
"... ..."
Classes and Recognition of Curve Contact Graphs
"... . Contact graphs are a special kind of intersection graphs of geometrical objects in which the objects are not allowed to cross but only to touch each other. Contact graphs of simple curves, and line segments as a special case, in the plane are considered. Various classes of contact graphs are intro ..."
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Cited by 7 (2 self)
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. Contact graphs are a special kind of intersection graphs of geometrical objects in which the objects are not allowed to cross but only to touch each other. Contact graphs of simple curves, and line segments as a special case, in the plane are considered. Various classes of contact graphs are introduced and the inclusions between them are described, also the recognition of the contact graphs is studied. As one of the main results, it is proved that the recognition of 3contact graphs is NPcomplete for planar graphs, while the same question for planar triangulations is polynomial. 1 Introduction The intersection graphs of geometrical objects have been extensively studied for their many practical applications. Formally the intersection graph of a set family M is defined as a graph G with the vertex set V (G) = M and the edge set E(G) = \Phi fA; Bg ` M j A 6= B; A " B 6= ; \Psi . Probably the first type studied were interval graphs, see [15],[1]; we may also mention other kinds ...
Contact Graphs of Curves
, 1995
"... Contact graphs are a special kind of intersection graphs of geometrical objects in which we do not allow the objects to cross but only to touch each other. Contact graphs of simple curves (and line segments as a special case) in the plane are considered. Several classes of contact graphs are introdu ..."
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Cited by 6 (4 self)
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Contact graphs are a special kind of intersection graphs of geometrical objects in which we do not allow the objects to cross but only to touch each other. Contact graphs of simple curves (and line segments as a special case) in the plane are considered. Several classes of contact graphs are introduced, their properties and inclusions between them are studied, and the maximal clique in relation with the chromatic number of contact graphs is considered. Also relations between planar and contact graphs are mentioned. Finally, it is proved that the recognition of contact graphs of curves (line segments) is NPcomplete (NPhard) even for planar graphs. 1 Introduction The intersection graphs of geometrical objects have been extensively studied for their many practical applications. Probably the first type studied were interval graphs (intersection graphs of intervals on a line), owing to their applications in biology, see [14],[1]. We may also mention other kinds of intersection graphs s...