Showing papers on "Planarity testing published in 1997"

An experimental comparison of four graph drawing algorithms

Abstract: In this paper we present an extensive experimental study comparing four general-purpose graph drawing algorithms. The four algorithms take as input general graphs (with no restrictions whatsoever on connectivity, planarity, etc.) and construct orthogonal grid drawings, which are widely used in software and database visualization applications. The test data (available by anonymous ftp) are 11,582 graphs, ranging from 10 to 100 vertices, which have been generated from a core set of 112 graphs used in “real-life” software engineering and database applications. The experiments provide a detailed quantitative evaluation of the performance of the four algorithms, and show that they exhibit trade-offs between “aesthetic” properties (e.g., crossings, bends, edge length) and running time.

Dummy fill patterns to improve interconnect planarity

Abstract: A method of improving the planarity of spin-on glass layers in semiconductor wafer processing is disclosed. Gaps in between active conductive traces on a trace layer of a semiconductor wafer that exceed a predetermined threshold distance are provided with dummy surfaces arranged in a micro-pattern in order to improve the planarity achieved in subsequently applied spin-on glass layers. In some embodiments, the predetermined threshold distance is greater than approximately 2 micrometers, as for example in the range of approximately 4.65 to 5 micrometers. In some applications, both the active conductive traces and the dummy surfaces are formed from a metallic material that is deposited in one single step with a dielectric layer being deposited over both the active conductive traces and the dummy surfaces prior to application of the spin-on glass layer.

On Triangulating Planar Graphs under the Four-Connectivity Constraint

Abstract: Triangulation of planar graphs under constraints is a fundamental problem in the representation of objects. Related keywords are graph augmentation from the field of graph algorithms and mesh generation from the field of computational geometry. We consider the triangulation problem for planar graphs under the constraint to satisfy 4-connectivity. A 4-connected planar graph has no separating triangles, i.e., cycles of length 3 which are not a face. We show that triangulating embedded planar graphs without introducing new separating triangles can be solved in linear time and space. If the initial graph had no separating triangle, the resulting triangulation is 4-connected. If the planar graph is not embedded, then deciding whether there exists an embedding with at most k separating triangles is NP-complete. For biconnected graphs a linear-time approximation which produces an embedding with at most twice the optimal number is presented. With this algorithm we can check in linear time whether a biconnected planar graph can be made 4-connected while maintaining planarity. Several related remarks and results are included.

Checking the convexity of polytopes and the planarity of subdivisions

Abstract: This paper considers the problem of verifying the correctness of geometric structures. In particular, we design simple optimal checkers for convex polytopes in two and higher dimensions, and for various types of planar subdivisions, such as triangulations, Delaunay triangulations, and convex subdivisions. Their performance is analyzed also in terms of the algorithmic degree, which characterizes the arithmetic precision required.

Five-membered cyclopalladated rings: Cambridge structural database analysis of geometrical parameters and ‘aromatic’ character

Abstract: The relationship between the planarity of five-membered cyclopalladated rings and calculated aromaticity indices has been investigated. Geometrical parameters for 126 crystal structures containing the fragments [Pd(NXCY)A 2 ] (X, Y = C or N; A = any ligand) were obtained from the Cambridge Structural Database. These were classified in terms of planarity by measuring the displacement of the N atom bonded to the Pd from the calculated mean plane formed by the other four atoms. The same classification was performed using the displacement of the Pd atom and the results compared. ‘Aromaticity’ indices for the five-membered cyclopalladated rings, V and HOMA, were also calculated based on experimental bond lengths and typically used in the analysis of five- or six-membered organic heterocycles. The correlation between calculated ‘aromaticity’ indices and planarity of the rings has been shown to be good. The stability and/or reactivity of the five-membered cyclopalladated rings suggested by the sequence of ‘aromaticity’ indices agrees well with the reactivity series of these compounds previously found for ligand-exchange reactions.

Planarity of N,N-Dimethylacetamide, (CH3)2NC(O)CH3

Abstract: The molecular structure of N,N-dimethylactamide, (CH3)2NC(O)CH3, was determined by gas electron diffraction (GED). A rigid model results in a vibrationally averaged structure with nonplanar configu...

A Short Proof of Kuratowski's Graph Planarity Criterion

Abstract: We present a new short combinatorial proof of the sufficiency part of the well-known Kuratowski's graph planarity criterion. The main steps are to prove that for a minor minimal non-planar graph G and any edge xy: (1) G-x-y does not contain -subgraph; (2) G-x-y is homeomorphic to the circle; (3) G is either K5 or Kf3;3g. c 1997 John Wiley & Sons, Inc.

Infinity and Planarity Test for Stereo Vision

Abstract: Introducing a mathematical model of noise in stereo images, we propose a new criterion for intelligent statistical inference about the scene we are viewing by using the geometric information criterion (geometric AIC). Using synthetic and real-image experiments, we demonstrate that a robot can test whether or not the object is located very far away or the object is a planar surface without using any knowledge about the noise magnitude or any empirically adjustable thresholds. key words: AIC, testing of hypotheses, model selection, stereo vision, in nity test, planarity test.

Efficient algorithms for finding disjoint paths in grids

Abstract: The reconfiguration problem on VLSI/WSI processor arrays in the presence of faulty processors can be stated as the following integral multi-source routing problem: Given a set of N nodes (faulty processors or sources) in am m x n rectangular grid where m, n {le} N, the problem to be solved is to connect the N nodes to distinct nodes at the grid boundary using a set of {open_quotes}disjoint{close_quotes} paths. This problem can be referred to as an escape problem which can be solved trivially in O(mnN) time. By exploiting all the properties of the network, planarity and regularity of a grid, integral flow, and unit capacity source/sink/flow, we can optimally compress the size of the grid from O(mn) to O({radical}mnN) and solve the problem in O(d{radical}mnN), where d is the maximum number of disjoint paths found, for both the edge-disjoint and vertex-disjoint cases. In the worst case, d, m, n are O(N) and the result is O(N{sup 2.5}). Note that this routing problem can also be solved with the same time complexity even if the disjoint paths have to be ended at another set of N nodes (sinks) in the grid instead of the grid boundary.

Quantum-chemical study of electronic properties of model spirooxazines

Abstract: Model spirooxazines were studied via ab initio quantum-chemical calculations. An anomeric interaction between nitrogen lone pair and antibonding orbital of the C spiro –O bond has been considered by Hartree–Fock SCF level of theory with 3-21G basis set. The extent of the n N –σ* CO interaction has been shown to depend on planarity of the nitrogen atom, being maximized with its planarity growth. Electronic properties and geometry parameters of model spirooxazines were considered by limited CI under 3-21G basis set. Excitation lengthens the C spiro –O bond and causes a significant polarization of total wave function, which results in increasing of the system dipole moment and transferring the electronic density from the O2 atom and the phenyl fragment onto the N13C14 bond in the case of spirooxazine and onto the NO 2 group in the case of nitro-spirooxazine. Elongation of the C–O bond under excitation is due to the NO 2 interaction in spirooxazines. CI calculation of the excited singlet state of a model chromene does not show elongation of this bond.

Planar topological routing

Abstract: We develop a simple linear time algorithm to determine if a collection of two-pin nets can be routed, topologically, in a plane (i.e., single layer). Experiments indicate that this algorithm is faster than the linear time algorithm of Marek-Sadowska and Tarng. Topological routability testing of a collection of multipin nets is shown to be equivalent to planarity testing, and a simple linear time algorithm is developed for the case when the collection of modules remains connected following the deletion of all nets with more than two pins.

Checking the Convexity of Polytopes and the Planarity of Subdivisions (Extended Abstract)

Abstract: This paper studies the problem of verifying the correctness of geometric structures. We design optimal checkers for convex polytopes in two and higher dimensions, and for various types of planar subdivisions, such as triangulations, Delaunay triangulations, and convex subdivisions. Our checkers are simpler and more general than the ones previously described in the literature. Their performance is studied also in terms of the degree, which characterizes the arithmetic precision required.

4,5,6,7,8,9‐Hexamethoxyindeno[1,2,3‐ij]isoquinoline

Abstract: The crystal and molecular structure of the title compound (I), has been determined to confirm the molecular conformation. We established by x-ray investigations the planarity of the polyaromatic ring-system, which is necessary for the formation of discotic mesophases.

Spin densities, conformations and rotational energy barriers around the C(O)N bond in acylaminoxyl (RC(O)N(O)R′; R,R′H,CH3) radicals: ab initio and density functional study

Abstract: Systematic ab initio (at UHF and UMP2 levels of theory) and density functional calculations (using non-local exchange and correlation functionals) were carried out for the C- and N-methyl substituted formylaminoxyl derivatives using a 6–31 + G∗ basis. For the formylaminoxyl radical, larger bases (up to 6–311 + + G∗∗, to calculate the geometry and Chipman's [6s,3p,2d] basis for spin densities) and CCSD and CCSD(T) types of wavefunction were also applied. The core part (the OCNO group) was found to be planar in their most stable (E) conformation in every case. Some small deviation from this planarity could be observed for their less stable (Z) conformation. The energy gap between these two stable conformers was in the range 4–9 kcal mol−1 depending on the compound and the level of theory, the most “crowded” example (N-acetyl-N-methylaminoxyl) exhibiting the largest energy difference. The rotational energy barriers around (O)CN(O) were less influenced by the methyl substitution. At the transition state, the aminoxyl moiety had lost its planarity. The (O)CN(O) bond was substantially longer in the transition state (by ~0.05 A) than in either of the energy minima, indicating a partial double-bond character in the latter cases. As a consequence of this delocalisation of the SOMO orbital, the relatively small aN hyperfine coupling constant (hfcc) can be exfplained by a smaller spin polarization at the N nucleus. However, one cannot neglect the effect of local planarity of the aminoxyl group on this hfcc as shown by the comparison between the rigid (keeping the radical center planar) and relaxed rotor approaches upon rotation around the (O)CN(O) bond.

On the planarity of infinite 2-complexes

Abstract: This paper is devoted to the problem of characterizing infinite planar polyhedra. We give two topological characterizations of planarity, and two others of proper planarity (embeddings with no accumulation points). We also give a combinatorial characterization of planarity, and proper planarity. In the case of compact polyhedra both results provide a weaker condition than that given by Gross and Rosen in [6].

Method and apparatus for measuring planarity

Abstract: PROBLEM TO BE SOLVED: To provide a method and apparatus for measuring the planarity of the bottom plate of paper feed cassette accurately and efficiently. SOLUTION: At the time of measuring the planarity of a bottom plate 13, a pin 21 fixed to an insertion means of air cylinder or the like is inserted into the holes 14a, 14b made through the arms 15a, 15b of bottom plate 13. The bottom plate 13 is supported rotatably, at the rear end, by means of the pin 21 and mounted, at the front end, on a plurality of digital displacement gauges 12a-12d. Under a state where the bottom plate 13 is supported rotatably by means of the pin 21, planarity at the front end of bottom plate 13 is measured by means of the digital displacement gauges 12a-12d.

Interactive, high-degree orthogonal graph layout

Abstract: A method is provided of producing an orthogonal graph drawing (409), while avoiding in large difficulties characteristic of the prior art. It is designed for nodes of arbitrarily high degree (433, 437, 441), thus avoiding imposing artificial structure. It does not require planarity. The method is incremental in nature, adding one node at a time (437). However, no specific demands are put on node ordering. In fact, nodes can be removed (441) and added (437) again, without destroying any invariant, if that is so desirable. Nodes are added in such a way that the edge lengths are kept small (419), and neighborhoods are preserved. The drawing grows equally in both horizontal and vertical directions.

Easy Planarity Testing for Ordered Sets

Abstract: The search for an efficient planarity-testing procedure for ordered sets is a longstanding problem. Even for small ordered sets it was often not easy do decide whether they have upward planar drawings or not. Only recently it has been proved that, in contrast with graphs, planarity for ordered sets is NP-complete [2], and a polynomial-time procedure has been found, on the other hand, to test whether a planar embedding of an acyclic digraph has a similar planar upward drawing [1]. The later shows that checking whether a triconnected ordered set is planar may be done also in polynomial-time. The procedure described in [1] uses to this end the Ford-Fulkerson algorithm solving the max-flow problem on a related network. In this note we present a direct method, which is simpler and for small ordered sets requires, in fact, almost no computations.

Planar graphs via squashing polyhedra and dynamic geometry

Abstract: Planarity is a concept that appears in most discrete mathematics courses (in particular, it is a syllabus item in all A level Discrete mathematics syllabuses). I have recently taught a unit in graph theory to a group of BEd students and, when I came to the concept of planarity, I decide to approach it in a different way – through the context of Cabri Geometre , a dynamic geometry package. In the article I would like to share some of my ideas and reflections on this approach to teaching this topic, paying particular attention to the role of Cabri Geometre in scaffolding the important ideas involved in an understanding of planarity.