Showing papers on "Planarity testing published in 2010"
TL;DR: This paper is the first one to investigate the problem of mining frequent subgraph patterns from uncertain graph data and uses efficient methods to determine whether a subgraph pattern can be output or not and a new pruning method to reduce the complexity of examining sub graph patterns.
Abstract: In many real applications, graph data is subject to uncertainties due to incompleteness and imprecision of data. Mining such uncertain graph data is semantically different from and computationally more challenging than mining conventional exact graph data. This paper investigates the problem of mining uncertain graph data and especially focuses on mining frequent subgraph patterns on an uncertain graph database. A novel model of uncertain graphs is presented, and the frequent subgraph pattern mining problem is formalized by introducing a new measure, called expected support. This problem is proved to be NP-hard. An approximate mining algorithm is proposed to find a set of approximately frequent subgraph patterns by allowing an error tolerance on expected supports of discovered subgraph patterns. The algorithm uses efficient methods to determine whether a subgraph pattern can be output or not and a new pruning method to reduce the complexity of examining subgraph patterns. Analytical and experimental results show that the algorithm is very efficient, accurate, and scalable for large uncertain graph databases. To the best of our knowledge, this paper is the first one to investigate the problem of mining frequent subgraph patterns from uncertain graph data.
170 citations
TL;DR: In this paper, the basic properties of a ring with nonzero identity are investigated and some characterization results regarding connectedness, chromatic index, diameter, girth, and planarity are given.
Abstract: Let R be a ring with nonzero identity. The unit graph of R, denoted by G(R), has its set of vertices equal to the set of all elements of R; distinct vertices x and y are adjacent if and only if x + y is a unit of R. In this article, the basic properties of G(R) are investigated and some characterization results regarding connectedness, chromatic index, diameter, girth, and planarity of G(R) are given. (These terms are defined in Definitions and Remarks 4.1, 5.1, 5.3, 5.9, and 5.13.)
112 citations
TL;DR: In this paper, a linearized version of group field theory is introduced, which can be viewed either as a group field over the additive group of a vector space or as an asymptotic expansion of any group field theories around the unit group element.
Abstract: We introduce a linearized version of group field theory. It can be viewed either as a group field theory over the additive group of a vector space or as an asymptotic expansion of any group field theory around the unit group element. We prove exact power-counting theorems for any graph of such models. For linearized colored models the power counting of any amplitude is further computed in terms of the homology of the graph. An exact power-counting theorem is also established for a particular class of graphs of the nonlinearized models, which satisfy a planarity condition. Examples and connections with previous results are discussed.
111 citations
17 Jan 2010
TL;DR: This work shows that the planarity question remains polynomial-time solvable, and implies that no dynamic programming is needed for a decision algorithm and that the elements of the decomposition can be processed independently.
Abstract: We study the following problem: Given a planar graph G and a planar drawing (embedding) of a subgraph of G, can such a drawing be extended to a planar drawing of the entire graph G? This problem fits the paradigm of extending a partial solution to a complete one, which has been studied before in many different settings. Unlike many cases, in which the presence of a partial solution in the input makes hard an otherwise easy problem, we show that the planarity question remains polynomial-time solvable. Our algorithm is based on several combinatorial lemmata which show that the planarity of partially embedded graphs meets the "on-cas" behaviour -- obvious necessary conditions for planarity are also sufficient. These conditions are expressed in terms of the interplay between (a) rotation schemes and containment relationships between cycles and (b) the decomposition of a graph into its connected, biconnected, and triconnected components. This implies that no dynamic programming is needed for a decision algorithm and that the elements of the decomposition can be processed independently.Further, by equipping the components of the decomposition with suitable data structures and by carefully splitting the problem into simpler subproblems, we improve our algorithm to reach linear-time complexity.Finally, we consider several generalizations of the problem, e.g. minimizing the number of edges of the partial embedding that need to be rerouted to extend it, and argue that they are NP-hard. Also, we show how our algorithm can be applied to solve related Graph Drawing problems.
97 citations
TL;DR: This work is motivated by applications in freeform architecture and leads to a discussion of fields of conjugate directions in surfaces, their singularities and indices, their optimization and their interactive modeling.
Abstract: We study the combined problem of approximating a surface by a quad mesh (or quad-dominant mesh) which on the one hand has planar faces, and which on the other hand is aesthetically pleasing and has evenly spaced vertices. This work is motivated by applications in freeform architecture and leads to a discussion of fields of conjugate directions in surfaces, their singularities and indices, their optimization and their interactive modeling. The actual meshing is performed by means of a level set method which is capable of handling combinatorial singularities, and which can deal with planarity, smoothness, and spacing issues.
78 citations
14 Mar 2010
TL;DR: This paper proves that the conflict graph used to model DPL conflicts in layout is a planar graph, and proposes a new face merging based framework which formulates DPL decomposition as a problem of pairing odd faces to simultaneously inimize the number of stitches generated and conflicts to eliminate.
Abstract: Double Patterning Lithography (DPL) is one of the few hopeful candidate solutions for the lithography for CMOS process beyond 45nm. DPL assigns the patterns less than a certain distance from each other on each layer onto two masks instead of one mask in traditional lithography. In this paper, we prove that the conflict graph used to model DPL conflicts in layout is a planar graph. Based on the planarity of the conflict graph, we propose a new face merging based framework which formulates DPL decomposition as a problem of pairing odd faces to simultaneously inimize the number of stitches generated and conflicts to eliminate. We employ partitioning and simplification techniques to reduce the problem size and use an O(n3) time maximum weighted matching algorithm to generate an optimal DPL decomposition.
56 citations
TL;DR: Results evidenced that noncovalent S...O interactions, which were postulated on the basis that the nonbonded distances between sulfur and oxygen atoms belonging to neighboring repeating units are slightly repulsive destabilizing the planar anti conformation, produce gain in aromaticity and favorable electrostatic interactions when the planarity is reached.
Abstract: The intramolecular interactions responsible for the planarity observed in poly(3,4-ethylenedioxythiophene) and small 3,4-ethylenedioxythiophene-containing oligomers have been investigated using quantum mechanical methods. Specifically, the relative influence of electron-donating effects, π-conjugation, and geometric restrictions induced by the cyclic substituent and attractive S···O intramolecular noncovalent interactions, which were proposed to be the most relevant factor for such planarity on the self-rigidification observed in these compounds, have been examined considering a wide number of model compounds. Results evidenced that noncovalent S···O interactions, which were postulated on the basis that the nonbonded distances between sulfur and oxygen atoms belonging to neighboring repeating units are significantly shorter than the sum of the van der Waals radii of sulfur and oxygen, are slightly repulsive destabilizing the planar anti conformation. In contrast, the latter arrangement is favored by the π...
36 citations
TL;DR: The restrictiveness of planarity on their complexities is explored and it is shown that both problems remain as hard as in the general case, that is, GapL- and P- complete.
Abstract: Viewing the computation of the determinant and the permanent of integer matrices as combinatorial problems on associated graphs, we explore the restrictiveness of planarity on their complexities and show that both problems remain as hard as in the general case, that is, GapL- and P- complete. On the other hand, both bipartite planarity and bimodal planarity bring the complexity of permanents down (but no further) to that of determinants. The permanent or the determinant modulo 2 is complete for ⊕L, and we show that parity of paths in a layered grid graph (which is bimodal planar) is also complete for this class. We also relate the complexity of grid graph reachability to that of testing existence/uniqueness of a perfect matching in a planar bipartite graph.
28 citations
21 Sep 2010
TL;DR: It is shown that using modern SAT and MIP solving approaches the authors can find optimal orderings for minimal crossing or minimal edge deletion for planarization on reasonably sized graphs.
Abstract: An important step in laying out hierarchical network diagrams is to order the nodes on each level. The usual approach is to minimize the number of edge crossings. This problem is NP-hard even for two layers when the first layer is fixed. Hence, in practice crossing minimization is performed using heuristics. Another suggested approach is to maximize the planar subgraph, i.e. find the least number of edges to delete to make the graph planar. Again this is performed using heuristics since minimal edge deletion for planarity is NP-hard.We show that using modern SAT and MIP solving approaches we can find optimal orderings for minimal crossing or minimal edge deletion for planarization on reasonably sized graphs. These exact approaches provide a benchmark for measuring quality of heuristic crossing minimization and planarization algorithms. Furthermore, we can straightforwardly extend our approach to minimize crossings followed by maximizing planar subgraph or vice versa; these hybrid approaches produce noticeably better layout then either crossing minimization or planarization alone.
23 citations
TL;DR: Several geometrical characteristics of 6-, 7-, 10-, 11- helicene have been studied: accuracy of the binary axis, distribution of atoms over three helices, bond distances and angles, planarity and interplanar angles of benzene rings, existence of new planar groups.
Abstract: Several geometrical characteristics of 6-, 7-, 10-, 11- helicene have been studied: accuracy of the binary axis, distribution of atoms over three helices, bond distances and angles, planarity and interplanar angles of benzene rings, existence of new planar groups. Two types of general models are suggested and discussed: the triple helix and the staircase model.
22 citations
TL;DR: In this paper, the role of integrability in certain aspects of N = 4 SYM which go beyond the planar spectrum is discussed, in particular in relation to nonplanar anomalous dimensions, multi-point functions and Maldacena-Wilson loops.
Abstract: We review the role of integrability in certain aspects of N=4 SYM which go beyond the planar spectrum. In particular, we discuss integrability in relation to non-planar anomalous dimensions, multi-point functions and Maldacena-Wilson loops.
TL;DR: The planarity theorems of MacLane and Whitney are extended to compact graph-like spaces and Whitney's Theorem for an identification space obtained from a graph in which no two vertices are joined by infinitely many edge-disjoint paths is applied.
Abstract: The planarity theorems of MacLane and Whitney are extended to compact graph-like spaces. This generalizes recent results of Bruhn and Stein (MacLane's Theorem for the Freudenthal compactification of a locally finite graph) and of Bruhn and Diestel (Whitney's Theorem for an identification space obtained from a graph in which no two vertices are joined by infinitely many edge-disjoint paths).
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TL;DR: A linear-time algorithm is given to test simultaneous planarity when the two graphs share a 2-connected subgraph and extended to the case of k planar graphs where each vertex [edge] is either common to all graphs or belongs to exactly one of them.
Abstract: Two planar graphs G1 and G2 sharing some vertices and edges are `simultaneously planar' if they have planar drawings such that a shared vertex [edge] is represented by the same point [curve] in both drawings. It is an open problem whether simultaneous planarity can be tested efficiently. We give a linear-time algorithm to test simultaneous planarity when the two graphs share a 2-connected subgraph. Our algorithm extends to the case of k planar graphs where each vertex [edge] is either common to all graphs or belongs to exactly one of them.
01 Dec 2010
TL;DR: In this paper, the authors characterize planar intersection graphs of ideals of a commutative ring with a constant number of vertices, i.e., 1, where vertices have vertices of the same type.
Abstract: In this paper we characterize planar intersection graphs of ideals of a commutative ring with 1.
15 Oct 2010
TL;DR: The author’s personal website is www.zusammenfassung.de.
Abstract: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Zusammenfassung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
21 Sep 2010
TL;DR: A new standard for visualizing graphs is studied: a monotone drawing is a straight-line drawing such that, for every pair of vertices, there exists a path that monotonically increases with respect to some direction.
Abstract: We study a new standard for visualizing graphs: A monotone drawing is a straight-line drawing such that, for every pair of vertices, there exists a path that monotonically increases with respect to some direction. We show algorithms for constructing monotone planar drawings of trees and biconnected planar graphs, we study the interplay between monotonicity, planarity, and convexity, and we outline a number of open problems and future research directions.
TL;DR: In this article, a linearized version of group field theory is introduced, which can be viewed either as a group field over the additive group of a vector space or as an asymptotic expansion of any group field theories around the unit group element.
Abstract: We introduce a linearized version of group field theory. It can be viewed either as a group field theory over the additive group of a vector space or as an asymptotic expansion of any group field theory around the unit group element. We prove exact power counting theorems for any graph of such models. For linearized colored models the power counting of any amplitude is further computed in term of the homology of the graph. An exact power counting theorem is also established for a particular class of graphs of the nonlinearized models, which satisfy a planarity condition. Examples and connections with previous results are discussed.
TL;DR: It is shown that for any data there exists a four-parameter system of interpolants and the one which preserves symmetry and planarity of the input data and which has the optimal approximation degree is identified.
TL;DR: A tree-decomposition based approach to solve non-local problems efficiently, such as Planar Hamiltonian Cycle in runtime 6^t^[email protected]?n^O^(^1^).
15 Dec 2010
TL;DR: In this paper, a linear-time algorithm is given to test simultaneous planarity when the two graphs share a 2-connected subgraph. But this algorithm is not applicable to the case of k-planar graphs, where each vertex is either common to all graphs or belongs to exactly one of them.
Abstract: Two planar graphs G 1 and G 2 sharing some vertices and edges are simultaneously planar if they have planar drawings such that a shared vertex [edge] is represented by the same point [curve] in both drawings. It is an open problem whether simultaneous planarity can be tested efficiently. We give a linear-time algorithm to test simultaneous planarity when the two graphs share a 2-connected subgraph. Our algorithm extends to the case of k planar graphs where each vertex [edge] is either common to all graphs or belongs to exactly one of them.
05 Jul 2010
TL;DR: This paper presents an algorithm that, given a set X of terminals in the plane, constructs a planar hop spanner with constant stretch factor for the Unit Disk Graph defined by X, and improves on previous constructions in the sense that it ensures the planarity of the whole spanner.
Abstract: The simplest model of a wireless network graph is the Unit Disk Graph (UDG): an edge exists in UDG if the Euclidean distance between its endpoints is ≤ 1. The problem of constructing planar spanners of Unit Disk Graphs with respect to the Euclidean distance has received considerable attention from researchers in computational geometry and ad-hoc wireless networks. In this paper, we present an algorithm that, given a set X of terminals in the plane, constructs a planar hop spanner with constant stretch factor for the Unit Disk Graph defined by X. Our algorithm improves on previous constructions in the sense that (i) it ensures the planarity of the whole spanner while previous algorithms ensure only the planarity of a backbone subgraph; (ii) the hop stretch factor of our spanner is significantly smaller.
TL;DR: This work considers the standard algorithm by Bertolazzi et al. to test the upward planarity of embedded digraphs and shows how to improve its running time from O(n+r^2) to O( n+ r^3^2), where r is the number of sources and sinks in the digraph.
13 Sep 2010
TL;DR: A method that allows local deformations of PQ-meshes (with square grid combinatorics) that makes it possible to modify a PZ-mesh while keeping all quadrilaterals planar through the whole process (without a minimization step).
Abstract: Planar quad-meshes (meshes with planar quadrilateral faces - PQ-meshes for short) are an important class of meshes (see e.g. Bobenko and Suris [2008]). Although they are often desirable in computer graphics - since planar quads can be rendered with out triangulating them - and architectual geometry (see Pottmann et al. [2007]) - because building with planar tiles ismore cost effective - modelling freeform surfaces with planar quadrilaterals is problematic (in fact in practical applications one deforms or subdivides PQ-meshes without obeying the planarity constraint and ensures it afterwards in a global optimization step). In this paper we present a method that allows local deformations of PQ-meshes (with square grid combinatorics) that makes it possible to modify a PQ-mesh while keeping all quadrilaterals planar through the whole process (without a minimization step). In principle the method allows for PQ-mesh subdivision as well.
The deformation scheme outlined in the following is a new approach and it is currently implemented in a prototype stage by Christian Hick. The general idea is that while moving a single vertex will generically destroy the planarity of the adiacent faces, moving the plane of a face and allowing its four points to adjust as necessary, has exactly enough freedom to generically allow for planarity of the central face as well as its neighbours.
08 Nov 2010
TL;DR: This work proposes a new method for modeling the vertical road profile from a disparity map based on a region-growing technique, which iteratively performs a least-squares fit of a B-spline curve to a region of selected points.
Abstract: Binocular vision combined with stereo matching algorithms can be used in vehicles to gather data of the spatial proximity. To utilize this data we propose a new method for modeling the vertical road profile from a disparity map. This method is based on a region-growing technique, which iteratively performs a least-squares fit of a B-spline curve to a region of selected points. We compare this technique to two variants of the v-disparity method using either an envelope function or a planarity assumption. Our findings are that the proposed road-modeling technique outperforms both variants of the v-disparity technique, for which the planarity assumption is slightly better than the envelope version.
Patent•
26 Aug 2010
TL;DR: In this paper, a planarity inspection of a film is performed by using an arithmetic processor to calculate the outside diameters of the film roll and their average, and a displacement as a difference between each outside diameter and the average, extracting the maximum positive displacement.
Abstract: PROBLEM TO BE SOLVED: To provide a planarity inspection apparatus of a film which noncontactly and instantaneously inspects the planarity of the film in a state that the film is rolled, and a planarity inspection method using the inspection apparatus SOLUTION: The planarity inspection apparatus includes a scanner 1 for scanning the outer circumference of the film roll 8 and an arithmetic processor 5 for processing a signal obtained from the scanner The scanner 1 includes a light emitting section 2 for irradiation with a laser light, a light receiving section 3 for receiving the laser light which is not blocked, and a signal conversion element 4 for outputting a numerical signal corresponding to the quantity of the received light Fz, and is configured so as to move relative to the film roll 8 in the width direction parallel to its axial line The arithmetic processor 5 calculates outside diameters of the film roll 8 and their average, calculates a displacement as a difference between each outside diameter and the average, and extracts the maximum positive displacement In the planarity inspection method, the quality of the film is determined by comparing the maximum positive displacement with a preset reference COPYRIGHT: (C)2010,JPO&INPIT
TL;DR: In this paper, a necessary and sufficient condition for the transformation graph G −++ to have crossing number one or two is established, where G is the graph with vertex set V(G) ∪ E(G), where the vertex x and y are joined by an edge.
Abstract: The transformation graph G –++ of G is the graph with vertex set V(G) ∪ E(G) in which the vertex x and y are joined by an edge if one of the following conditions holds: (i) x, y ∈ V(G) and x and y are not adjacent in G, (ii) x, y ∈ E(G) and x and y are adjacent in G, (iii) one of x and y is in V(G) and the other is in E(G), and they are incident in G. In this paper we present characterizations of graphs whose transformation graphs G –++ are eulerian, outerplanar, maximal outerplanar or minimally nonouterplanar. Further we establish a necessary and sufficient condition for the transformation graphs G –++ to have crossing number one or two.
10 Feb 2010
TL;DR: This paper provides characterizations for the class of directed trees that admit a switch-regular upward planar drawing and describes an optimal linear-time testing and embedding algorithm for this problem.
Abstract: Upward planar drawings of digraphs are crossing free drawings where all edges flow in the upward direction. The problem of deciding whether a digraph admits an upward planar drawing is called the upward planarity testing problem, and it has been widely studied in the literature. In this paper we investigate a new version of this problem: Deciding whether a digraph admits a switch-regular upward planar drawing, i.e., an upward planar drawing with specific topological properties. Switch-regular upward planar drawings have applications in the design of efficient checkers and in the design of effective compaction heuristics. We provide characterizations for the class of directed trees that admit a switch-regular upward planar drawing. Based on these characterizations we describe an optimal linear-time testing and embedding algorithm.
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TL;DR: In this paper, the authors focus on the class of planar Ising models, for which exact inference is tractable using techniques of statistical physics, and propose a simple greedy algorithm for learning the best Ising model to approximate an arbitrary collection of binary random variables (possibly from sample data).
Abstract: Inference and learning of graphical models are both well-studied problems in statistics and machine learning that have found many applications in science and engineering. However, exact inference is intractable in general graphical models, which suggests the problem of seeking the best approximation to a collection of random variables within some tractable family of graphical models. In this paper, we focus on the class of planar Ising models, for which exact inference is tractable using techniques of statistical physics. Based on these techniques and recent methods for planarity testing and planar embedding, we propose a simple greedy algorithm for learning the best planar Ising model to approximate an arbitrary collection of binary random variables (possibly from sample data). Given the set of all pairwise correlations among variables, we select a planar graph and optimal planar Ising model defined on this graph to best approximate that set of correlations. We demonstrate our method in simulations and for the application of modeling senate voting records.
01 Aug 2010
TL;DR: The goal of this work is to maintain routing guarantees while disturbing the network graph as little as possible, a new starting point emerges from which to build rich distributed protocols in the spirit of protocols such as CLDP and GDSTR.
Abstract: In this report we investigate the limits of routing according to leftor right-hand rule (LHR). Using LHR, a node upon receipt of a message will forward to the neighbour that sits next in counter-clockwise order in the network graph. When used to recover from greedy routing failures, LHR guarantees success if implemented over planar graphs. This is often referred to as face or geographic routing. In the current body of knowledge it is known that if planarity is violated then LHR is guaranteed only to eventually return to the point of origin. Our work seeks to understand why a non-planar environment stops LHR from making delivery guarantees. Our investigation begins with an analysis to enumerate all node configurations that cause intersections. A trace over each configuration reveals that LHR is able to recover from all but a single case, the ‘umbrella’ configuration so named for its appearance. We use this information to propose the Prohibitive Link Detection Protocol (PDLP) that can guarantee delivery over non-planar graphs using standard face-routing techniques. As the name implies, the protocol detects and circumvents the ‘bad’ links that hamper LHR. The goal of this work is to maintain routing guarantees while disturbing the network graph as little as possible. In doing so, a new starting point emerges from which to build rich distributed protocols in the spirit of protocols such as CLDP and GDSTR.