Showing papers on "Planarity testing published in 1999"

The Colin de Verdière graph parameter

Abstract: In 1990, Y. Colin de Verdi ere introduced a new graph parameter (G), based on spectral properties of matrices associated with G. He showed that (G) is monotone under taking minors and that planarity of G is characterized by the inequality (G) 3. Recently Lov asz and Schrijver showed that linkless embeddability of G is characterized by the inequality (G) 4. In this paper we give an overview of results on (G) and of techniques to handle it. Contents

Planar-Tetracoordinate Carbon in a Neutral Saturated Hydrocarbon: Theoretical Design and Characterization.

Abstract: Exact planarity at the central carbon atom is achieved, according to molecular orbital calculations, in the strained polycyclic cage hydrocarbon dimethanospiro[2.2]octaplane (see structure). There are no glaringly long C-C bonds, which might have reflected inherent instability in this molecule that is yet to be synthesized.

Planarity of para Hexaphenyl

Abstract: We present experimental and theoretical findings on the geometry of polycrystalline para hexaphenyl via Raman scattering. The planarity of the molecule is affected by hydrostatic pressure and temperature. Our studies indicate that the potential energy curve which governs the torsional motion between neighboring phenyl rings is ``W'' shaped. We determine the activation energy to promote the molecule from a nonplanar to a planar state to be 0.04 eV, in good agreement with our quantum chemical calculations. From the relative intensities of the $1280{\mathrm{cm}}^{\ensuremath{-}1}$ to the $1220{\mathrm{cm}}^{\ensuremath{-}1}$ Raman modes we show that high pressure planarizes the molecules, modifying the ``W''-shaped potential energy curve to a ``U''-shaped one.

Map Graphs

Abstract: We consider a modified notion of planarity, in which two nations of a map are considered adjacent when they share any point of their boundaries (not necessarily an edge, as planarity requires) Such adjacencies define a map graph We give an NP characterization for such graphs, and a cubic time recognition algorithm for a restricted version: given a graph, decide whether it is realized by adjacencies in a map without holes, in which at most four nations meet at any point

Level Planar Embedding in linear Time

Abstract: In a level directed acyclic graph G = (V;E) the vertex set V is partitioned into k ≤ |V | levels V1; V2... Vk such that for each edge (u, v) ∈ E with u ∈ Vi and v ∈; Vj we have i < j. The level planarity testing problem is to decide if G can be drawn in the plane such that for each level Vi, all v ∈ Vi are drawn on the line li = {(x, k - i) | x ∈ ℝ}, the edges are drawn monotonically with respect to the vertical direction, and no edges intersect except at their end vertices. In order to draw a level planar graph without edge crossings, a level planar embedding of the level graph has to be computed. Level planar embeddings are characterized by linear orderings of the vertices in each Vi (1 ≤ i ≤ k). We present an O(|V |) time algorithm for embedding level planar graphs. This approach is based on a level planarity test by Junger, Leipert, and Mutzel [6].

The electronic structure and chemical bonding of hypermetallic Al5C by ab initio calculations and anion photoelectron spectroscopy

Abstract: The chemical structure and bonding of the hypermetallic Al5C and Al5C− species have been studied by photoelectron spectroscopy and ab initio calculations. Both Al5C (C2v, 2A1) and Al5C− (C2v, 1A1) are found to have planar structures that can be related to that of the planar square Al4C− by adding one Al+ ion or one Al atom to an edge of the square. The planarity of Al5C and Al5C− can be explained in terms of the structure of their highest occupied molecular orbitals which are ligand five-center one- or two-electron bonding MO, respectively, similar to the orbital responsible for the planarity of Al4C−. Four peaks were observed in the photoelectron spectra of Al5C− with vertical binding energies of 2.67, 2.91, 3.19, and 4.14 eV which compare well with the 2.68, 2.96, 3.27, and 4.35 eV calculated by the Green function method [OVGF/6-311+G(2df )]. The excellent agreement between the calculated and experimental electron affinity and excitation energies allow us to completely elucidate the geometrical and elec...

The Vertex-Exchange Graph: A New Concept for Multi-level Crossing Minimisation

Abstract: In this paper we consider the problems of testing a multi- level graph for planarity and laying out a multi-level graph.We introduce a new abstraction that we call a vertex-exchange graph. We demonstrate how this concept can be used to solve these problems by providing clear and simple algorithms for testing a multi-level graph for planarity and laying out a multi-level graph when planar.We also show how the concept can be used to solve other problems relating to multi-level graph layout.

Absolute measurement of planarity with Fritz's method: uncertainty evaluation.

Abstract: Fritz’s method [Opt. Eng.23, 379 (1984)] of using Zernike polynomials to assess the absolute planarity of test plates is revisited. A refinement is described that takes into account the data decorrelation that appears in experiments. An uncertainty balance is defined by propagation of error contributions through the steps of the method. The resultant measuring procedure is demonstrated on a data set from experiments, and a nanometer level of uncertainty is achieved.

Separator-Based Sparsification II: Edge and Vertex Connectivity

Abstract: We consider the problem of maintaining a dynamic planar graph subject to edge insertions and edge deletions that preserve planarity but that can change the embedding. We describe algorithms and data structures for maintaining information about 2- and 3-vertex-connectivity, and 3- and 4-edge-connectivity in a planar graph in O(n1/2) amortized time per insertion, deletion, or connectivity query. All of the data structures handle insertions that keep the graph planar without regard to any particular embedding of the graph. Our algorithms are based on a new type of sparsification combined with several properties of separators in planar graphs.

Fully dynamic planarity testing with applications

Abstract: This paper introduces compressed certificates for planarity, biconnectivity and triconnectivity in planar graphs, and proves many structural properties of certificates in planar graphs. As an application of our compressed certificates, we develop efficient dynamic planar algorithms. In particular, we consider the following three operations on a planar graph G: (i) insert an edge if the resultant graph remains planar; (ii) delete an edge; and (iii) test whether an edge could be added to the graph without violating planarity. We show how to support each of the above operations in O(n2/3) time, where n is the number of vertices in the graph. The bound for tests and deletions is worst-case, while the bound for insertions is amortized. This is the first algorithm for this problem with sub-linear running time, and it affirmatively answers a question posed in Epstein et al. [1992]. We use our compressed certificates for biconnectivity and triconnectivity to maintain the biconnected and triconnected components of a dynamic planar graph. The time bounds are the same: O(n2/3) worst-case time per edge deletion, O(n2/3) amortized time per edge insertion, and O(n2/3) amortized time per edge insertion, and O(n2/3)worst-case time to check whether two vertices are either biconnected or triconnected.

Prealigner and planarity teaching station

Abstract: A planarity teaching station (14) includes one or more proximity sensors (42, 62, 72) to reflect light off a surface to determine a plane in which a substrate is positioned by a robotic arm (20). The plane of the substrate is then automatically adjusted by changing a Z-axis of a universal tiltable robot base or in another manner, based on the data provided by the planarity teaching station (14). The adjustment of the substrate or end effector plane (22) allows substrates to be removed from and delivered to various cassettes (10) and workstations (12) of a substrate processing system without damage to the substrate caused by end effector misalignment.

Planarity-Preserving Clustering and Embedding for Large Planar Graphs

Abstract: In this paper we present a novel approach for cluster-based drawing of large planar graphs that maintains planarity. Our technique works for arbitrary planar graphs and produces a clustering which satisfies the conditions for compound-planarity (c-planarity). Using the clustering, we obtain a representation of the graph as a collection of O(log n) layers, where each succeeding layer represents the graph in an increasing level of detail. At the same time, the difference between two graphs on neighboring layers of the hierarchy is small, thus preserving the viewer’s mental map. The overall running time of the algorithm is O(n log n), where n is the number of vertices of graph G.

Inter-Ring Torsions in N-Phenylmaleimide and Its o-Halo Derivatives: An Experimental and Computational Study

Abstract: Structures of N-phenylmaleimide and its o-halophenyl derivatives have been determined in the solid state and show the angle between the phenyl and pyrolinyl ring planes to vary from 49.5° to 83.9° with increasing values for compounds with the larger ortho halophenyl substituents (H < F ≲ Cl ≲ Br < I). Experimental torsions and trends in the series are supported by semiempirical AM1 and ab initio SCF, DFT, and MP2 calculations. Calculations (AM1) on N-phenylmaleimide modeling the torsional deformation between the rings show that the barrier to planarity has a lower energy than that through a perpendicular conformation. In its o-halo derivatives, molecular planarity is not possible, and torsional deformation proceeds through the perpendicular conformation with diminishing, possibly vanishing, barriers with increasing halogen size. For chloro, bromo, and iodo derivatives, twisted ground-state molecular conformations reside in broad minima essentially centered around the perpendicular conformations. The unusu...

Graph Planarity and Related Topics

Abstract: This compendium is the result of reformatting and minor editing of the author’s transparencies for his talk delivered at the conference. The talk covered Euler’s Formula, Kuratowski’s Theorem, linear planarity tests, Schnyder’s Theorem and drawing on the grid, the two paths problem, Pfaffian orientations, linkless embeddings, and the Four Color Theorem.

Method of and apparatus for measuring planarity of metal strip

Abstract: The planarity of a strip, especially a metal strip is measured in a measurement zone in which the tension of the strip has been reduced in a targeted way by braking the strip. The braking force should be practically equal to the strip tension.

On the planarity errors of the hologram of the CATR

Abstract: Conventional compact antenna test ranges (CATR) are based on reflectors, the problem of which is a very tight requirement on the surface accuracy. A hologram is a transmission type of device, thus the required surface accuracy of the hologram is not so stringent, and the manufacturing of the hologram is much less expensive than that of the reflector. The effect of the planarity errors of the hologram on the quiet-zone field are simulated (with a method based on the two dimensional finite difference time domain method (FDTD) and physical optics), and the simulation results are compared with a simple theoretical path length approximation.

Quasi-upward planarity

Abstract: AbstractAn upward planar drawing of a directed graph (digraph) is a planar drawing such that all the edges are represented by curves monotonically increasing in the vertical direction. In this paper we introduce and study the concept of quasi-upward planarity. Quasi-upward planarity allows us to extend the upward planarity theory to a large class of digraphs including digraphs that also have directed cycles. First, we characterize the digraphs that have a quasi-upward planar drawing. Second, we give a polynomial time algorithm for computing ``optimal'' quasi-upward planar drawings within a given planar embedding. Further, we apply branch and bound techniques to the problem of computing optimal quasi-upward planar drawings, considering all possible planar embeddings. The paper also contains experimental results about the proposed techniques.

Far-Infrared and Raman Spectra of the Ring-Puckering Vibration of 2,3-Dihydrothiophene. One- and Two-Dimensional Potential Energy Surfaces and the Barrier to Planarity

Abstract: The vapor-phase far-infrared and Raman spectra of 2,3-dihydrothiophene have been recorded and analyzed. The infrared spectra show more than fifty transition frequencies corresponding to ΔvP = 1, 2, 3, 4, and 5 transitions. The ring-puckering energy levels were determined for both the ring-twisting ground and excited states. Both one- and two-dimensional potential energy functions, which fit the observed data very well, were determined. The barrier to planarity was determined to be 430 cm-1 from the one-dimensional model and 435 cm-1 for the two-dimensional model. The experimental dihedral angle of puckering is 31°, while an ab initio calculation predicts 29°. The magnitude of the interaction constant between the puckering and twisting was found to be 1.67 × 105 cm-1/A4, similar to the values determined for related molecules.

A Study of the Planarity by Sti Cmp Erosion Modeling

Abstract: In this work, we propose a new equation that predicts the planarity as a function of active pattern density, initial step height, selectivity between gapfilled oxide and silicon nitride and over CMP amounts. In order to achieve highly planarized STI surface, uniform active density, reduced initial step height, minimization of over CMP amounts and high selective slurry were required. Our new equation was applied to the 0.18um graded CPU devices’ STI CMP to enhance planarity and these parameters were evaluated quantitatively. It is concluded that the model suggested is useful in predicting CMP planarity

Projective decomposition of planar facets

Abstract: Since for recognition tasks it is known that planar invariants are more easily obtained than others, decomposing a scene in terms of planar parts becomes very interesting. This paper presents a new approach to finding the projections of planar surfaces in a pair of images. For this task we introduce the facet concept defined by linked edges (chains) and corners. We use collineations as projective information to match and verify their planarity. Collineations are constrained by the fundamental matrix information and a Kalman filter approach is used to refine its computation. Furthermore the Kalman filter is used to enrich the number of detected facets by finding coplanar facets.

Conjugation in Three-Dimensional Curvature of p-Conjugation Systems

Abstract: The semiempirical AM1-calculated cubic hyperpolarizability is found to be surprisingly low in C , which is in agreement with recent experimental results In 60 the past, 10 orders of magnitude discrepancy has existed in the literature for the cubic hyperpolarizabilities of C Comparing some selected groups of polyaromatic 60 hydrocarbons, we show that linearity and planarity are two dominating factors in the second hyperpolarizability in polyaromatic rings Therefore, the observations that aromatic rings on C lose their linearity and planarity account for the low second 60 hyperpolarizability Furthermore, the planarity defined mathematically can be served as a scale of conjugation in p-conjugation systems By using biphenyl as a reference, the cubic hyperpolarizability of C can be estimated by the scaling of conjugation Q 1999

Effect of the control of global planarity of intermetal dielectric layers on the lithographic process window

Abstract: The sensitivity of lithographic process window to global planarity of the inter metal dielectric layers is established in this work. The inter metal dielectric layers, between the metal layers, were prepared by utilizing the H 2 O 2 /SiH 4 chemistry known as the 'Advanced Planarity Layer (APL)'. Four degrees of global planarity were tested within the APL process window, utilizing different H 2 O 2 stabilization pressures. SEM cross sections were used to determine the degree of planarity in the CMOS product and at lithographic test structures. The lithographic process window and the effect of the stepper leveling system were defined for typical high and low topographies. The results how a strong link between the lithographic process window to degree of global planarity of the APL. Good global planarity enlarged depth of focus and energy latitude, allowing a wider lithographic process window. Also, in cases of improved APL planarity, the stepper leveling system had only a limited contribution to a lithographic process window. This control over the global planarity of the inter metal dielectric layers and the wide lithographic process window that results eliminate the need for CMP at 0.5 (mu) technology.

Measuring the absolute planarity of test plates with a modified Fritz's method

Abstract: The Fritz's method using Zernike polynomials to assess the absolute planarity of test plates is revisited. Such method is based on four interferometric measurements, which are assumed perfectly correlated. In experiments, due to several instability sources, the data set is missing perfect correlation. Modifications of the Fritz's method are here presented, taking into account the residual uncorrelation of the data; such modified approach is demonstrated on a data set from experiments, achieving nanometer uncertainty level.