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166
Property Testing in Bounded Degree Graphs
 Algorithmica
, 1997
"... We further develop the study of testing graph properties as initiated by Goldreich, Goldwasser and Ron. Whereas they view graphs as represented by their adjacency matrix and measure distance between graphs as a fraction of all possible vertex pairs, we view graphs as represented by boundedlength in ..."
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Cited by 119 (36 self)
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We further develop the study of testing graph properties as initiated by Goldreich, Goldwasser and Ron. Whereas they view graphs as represented by their adjacency matrix and measure distance between graphs as a fraction of all possible vertex pairs, we view graphs as represented by boundedlength incidence lists and measure distance between graphs as a fraction of the maximum possible number of edges. Thus, while the previous model is most appropriate for the study of dense graphs, our model is most appropriate for the study of boundeddegree graphs. In particular, we present randomized algorithms for testing whether an unknown boundeddegree graph is connected, kconnected (for k ? 1), planar, etc. Our algorithms work in time polynomial in 1=ffl, always accept the graph when it has the tested property, and reject with high probability if the graph is fflaway from having the property. For example, the 2Connectivity algorithm rejects (w.h.p.) any Nvertex ddegree graph for which more ...
Connectivity and Inference Problems for Temporal Networks
 J. Comput. Syst. Sci
, 2000
"... Many network problems are based on fundamental relationships involving time. Consider, for example, the problems of modeling the flow of information through a distributed network, studying the spread of a disease through a population, or analyzing the reachability properties of an airline timetable. ..."
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Cited by 49 (3 self)
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Many network problems are based on fundamental relationships involving time. Consider, for example, the problems of modeling the flow of information through a distributed network, studying the spread of a disease through a population, or analyzing the reachability properties of an airline timetable. In such settings, a natural model is that of a graph in which each edge is annotated with a time label specifying the time at which its endpoints “communicated. ” We will call such a graph a temporal network. To model the notion that information in such a network “flows ” only on paths whose labels respect the ordering of time, we call a path timerespecting if the time labels on its edges are nondecreasing. The central motivation for our work is the following question: how do the basic combinatorial and algorithmic properties of graphs change when we impose this additional temporal condition? The notion of a path is intrinsic to many of the most fundamental algorithmic problems on graphs; spanning trees, connectivity, flows, and cuts are some examples. When we focus on timerespecting paths in place of arbitrary paths, many of these problems acquire a character that is different from the
Shallow Excluded Minors and Improved Graph Decompositions
, 1994
"... In this paper we introduce the notion of the limiteddepth minor exclusion and show that graphs that exclude small limiteddepth minors have relatively small separators. In particular, we prove that for any graph that excludes K h as a depth l minor, we can find a separator of size O(lh 2 log n n=l) ..."
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Cited by 34 (2 self)
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In this paper we introduce the notion of the limiteddepth minor exclusion and show that graphs that exclude small limiteddepth minors have relatively small separators. In particular, we prove that for any graph that excludes K h as a depth l minor, we can find a separator of size O(lh 2 log n n=l). This, in turn, implies that any graph that excludes K h as a minor has an O(h p n log n)sized separator, improving the result of Alon, Seymour, and Thomas for the case where h AE p log n. We show that the ddimensional simplicial graphs with constant aspect ratio, defined by Miller and Thurston, exclude K h minors of depth L for h = \Omega\Gamma L d\Gamma1 ) when d is a constant. These graphs arise in finite element computations. Our proof of separator existence is constructive and gives an algorithm to find the tcutcovers decomposition, introduced by Kaklamanis, Krizanc, and Rao, in graphs that exclude small depth minors. This has two interesting implications. F...
ThreeDimensional Orthogonal Graph Drawing
, 2000
"... vi Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Tables . . . . . . . . . . . . ..."
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Cited by 27 (10 self)
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vi Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii List of Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv I Orthogonal Graph Drawing 1 1
Every minorclosed property of sparse graphs is testable
, 2007
"... Suppose G is a graph of bounded degree d, and one needs to remove ɛn of its edges in order to make it planar. We show that in this case the statistics of local neighborhoods around vertices of G is far from the statistics of local neighborhoods around vertices of any planar graph G ′. In fact, a sim ..."
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Cited by 25 (3 self)
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Suppose G is a graph of bounded degree d, and one needs to remove ɛn of its edges in order to make it planar. We show that in this case the statistics of local neighborhoods around vertices of G is far from the statistics of local neighborhoods around vertices of any planar graph G ′. In fact, a similar result is proved for any minorclosed property of bounded degree graphs. As an immediate corollary of the above result we infer that many well studied graph properties, like being planar, outerplanar, seriesparallel, bounded genus, bounded treewidth and several others, are testable with a constant number of queries. None of these properties was previously known to be testable even with o(n) queries. 1
RNA Structures with PseudoKnots  GraphTheoretical and Combinatorial Properties
, 1997
"... Secondary structures of nucleic acids are a particularly interesting class of contact structures. Many important RNA molecules contain pseudoknots, which are excluded explicitly by the definition of secondary structures. We propose here a generalization of secondary structures that incorporates &quo ..."
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Cited by 24 (7 self)
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Secondary structures of nucleic acids are a particularly interesting class of contact structures. Many important RNA molecules contain pseudoknots, which are excluded explicitly by the definition of secondary structures. We propose here a generalization of secondary structures that incorporates "nonnested" pseudoknots. We also introduce a measure for the complexity of more general contact structures in terms of the chromatic number of their intersection graph. We show that RNA structures without nested pseudoknots form a special class of planar graphs. Upper bounds on their number are derived, showing that there are fewer different structures than sequences. 1. Introduction Presumably the most important problem and the greatest challenge in present day theoretical biophysics is deciphering the code that transforms sequences of biopolymers into spatial molecular structures. A sequence is properly visualized as a string of symbols which together with the environment encodes the molecul...
Spatial Logic and the Complexity of Diagrammatic Reasoning
 MACHINE GRAPHICS AND VISION
, 1997
"... Researchers have sought to explain the observed "efficacy" of diagrammatic reasoning (DR) via the notions of "limited abstraction" and inexpressivity [17, 20]. We argue that application of the concepts of computational complexity to systems of diagrammatic representation is neces ..."
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Cited by 22 (2 self)
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Researchers have sought to explain the observed "efficacy" of diagrammatic reasoning (DR) via the notions of "limited abstraction" and inexpressivity [17, 20]. We argue that application of the concepts of computational complexity to systems of diagrammatic representation is necessary for the evaluation of precise claims about their efficacy. We show here how to give such an analysis. Centrally, we claim that recent formal analyses of diagrammatic representations (DRs) (eg: [14]) fail to account for the ways in which they employ spatial relations in their representational work. This focus raises some problems for the expressive power of graphical systems, related to the topological and geometrical constraints of the medium. A further idea is that some diagrammatic reasoning may be analysed as a variety of topological inference [15]. In particular, we show how reasoning in some diagrammatic systems is of polynomial complexity, while reasoning in others is NP hard. A simple case study i...
RNA Structures with Pseudoknots
, 1997
"... i Abstract Secondary structures of nucleic acids are a particularly interesting class of contact structures. Many important RNA molecules,however contain pseudoknots, which are excluded explicitly by the definition of secondary structures. We propose here a generalization of secondary structures th ..."
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Cited by 18 (1 self)
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i Abstract Secondary structures of nucleic acids are a particularly interesting class of contact structures. Many important RNA molecules,however contain pseudoknots, which are excluded explicitly by the definition of secondary structures. We propose here a generalization of secondary structures that incorporates "nonnested" pseudoknots. We also introduce a measure for the complexity of more general contact structures in terms of the chromatic number of their intersection graph. We show that RNA structures without nested pseudoknots form a special class of planar graphs, the so called "bisecondary structures". Upper bounds on their number are derived, showing that there are fewer different structures than sequences. An energy function capable of dealing with bisecondary structures was implemented into a generalized kinetic folding algorithm. Sterical hindrances involved in pseudoknot formation are taken into account with the help of two simplifications: stacked regions are viewed ...
Semantical Foundations of Spatial Logics
, 1996
"... We explore several "spatial" logics and investigate their credentials as logics of space or of spatial objects and relations. A semantical adequacy criterion for spatial logics is developed, according to which a logic is spatial only if consistent theories in that logic are realizable in ..."
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Cited by 18 (4 self)
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We explore several "spatial" logics and investigate their credentials as logics of space or of spatial objects and relations. A semantical adequacy criterion for spatial logics is developed, according to which a logic is spatial only if consistent theories in that logic are realizable in a standard model of space. Various (socalled) spatial logics are shown not to satisfy this criterion. In effect, these observations amount to incompleteness results for classes of spatial logics, because they show that consistent sets of formulae in these logics have no models of the intended sort. In addition, we present a complete axiomatization of space; a modal logic of connected regions of the plane. 1 MOTIVATION In recent years there has been much interest in logics of space (eg: [Randell et al., 1992, Gotts et al., 1996, Asher and Vieu, 1995, Bennett, 1995]). Different research programmes have developed from work (eg: [Tarski, 1956, Whitehead, 1929, Clarke, 1981, Clarke, 1985]) on logic and m...