Straight-Line Grid Drawings of Label-Constrained Outerplanar Graphs with O(n logn) Area (Extended Abstract)

2009 ◽  
pp. 310-321
Author(s):  
Md. Rezaul Karim ◽  
Md. Jawaherul Alam ◽  
Md. Saidur Rahman
Keyword(s):  
Outerplanar Graphs ◽  
Straight Line

SIMULTANEOUS EMBEDDING OF OUTERPLANAR GRAPHS, PATHS, AND CYCLES

2007 ◽  
Vol 17 (02) ◽  
pp. 139-160 ◽  
Author(s):  
EMILIO DI GIACOMO ◽  
GIUSEPPE LIOTTA
Keyword(s):  
Planar Graphs ◽  
The Other ◽  
Simple Path ◽  
Outerplanar Graph ◽  
Outerplanar Graphs ◽  
Line Segments ◽  
Straight Line ◽  
Simultaneous Embedding

Let G1 and G2 be two planar graphs having some vertices in common. A simultaneous embedding of G1 and G2 is a pair of crossing-free drawings of G1 and G2 such that each vertex in common is represented by the same point in both drawings. In this paper we show that an outerplanar graph and a simple path can be simultaneously embedded with fixed edges such that the edges in common are straight-line segments while the other edges of the outerplanar graph can have at most one bend per edge. We then exploit the technique for outerplanar graphs and paths to study simultaneous embeddings of other pairs of graphs. Namely, we study simultaneous embedding with fixed edges of: (i) two outerplanar graphs sharing a forest of paths and (ii) an outerplanar graph and a cycle.

scholarly journals Straight-Line Grid Drawings of Label-Constrained Outerplanar Graphs with O(n log n) Area

2011 ◽  
Vol 15 (3) ◽  
pp. 437-456
Author(s):  
Md. Rezaul Karim ◽  
Md. Jawaherul Alam ◽  
Md. Saidur Rahman
Keyword(s):  
Outerplanar Graphs ◽  
Straight Line

ON MINIMUM AREA PLANAR UPWARD DRAWINGS OF DIRECTED TREES AND OTHER FAMILIES OF DIRECTED ACYCLIC GRAPHS

2008 ◽  
Vol 18 (03) ◽  
pp. 251-271 ◽  
Author(s):  
FABRIZIO FRATI
Keyword(s):  
Bipartite Graphs ◽  
Directed Acyclic Graphs ◽  
Outerplanar Graph ◽  
Minimum Area ◽  
Outerplanar Graphs ◽  
Acyclic Graphs ◽  
Straight Line ◽  
Optimal Values ◽  
Area Occupation ◽  
Planar Drawing

It has been shown that there exist planar digraphs that require exponential area in every upward straight-line planar drawing. On the other hand, upward poly-line planar drawings of planar graphs can be realized in Θ(n2) area. In this paper we consider families of DAGs that naturally arise in practice, like DAGs whose underlying graph is a tree (directed trees), is a bipartite graph (directed bipartite graphs), or is an outerplanar graph (directed outerplanar graphs). Concerning directed trees, we show that optimal Θ(n log n) area upward straight-line/poly-line planar drawings can be constructed. However, we prove that if the order of the neighbors of each node is assigned, then exponential area is required for straight-line upward drawings and quadratic area is required for poly-line upward drawings, results surprisingly and sharply contrasting with the area bounds for planar upward drawings of undirected trees. After having established tight bounds on the area requirements of planar upward drawings of several families of directed trees, we show how the results obtained for trees can be exploited to determine asymptotic optimal values for the area occupation of planar upward drawings of directed bipartite graphs and directed outerplanar graphs.

Microdensitometer—computer correlation analysis of ultrastructural periodicity

1978 ◽  
Vol 36 (2) ◽  
pp. 32-33
Author(s):  
D.R. Ensor ◽  
C.G. Jensen ◽  
J.A. Fillery ◽  
R.J.K. Baker
Keyword(s):  
Correlation Analysis ◽  
Background Noise ◽  
Computer Control ◽  
Optical Diffraction ◽  
Screw Thread ◽  
Straight Line ◽  
Computer Storage ◽  
Correlation Process ◽  
Electron Microscope Image ◽  
Fixed Beam

Because periodicity is a major indicator of structural organisation numerous methods have been devised to demonstrate periodicity masked by background “noise” in the electron microscope image (e.g. photographic image reinforcement, Markham et al, 1964; optical diffraction techniques, Horne, 1977; McIntosh,1974). Computer correlation analysis of a densitometer tracing provides another means of minimising "noise". The correlation process uncovers periodic information by cancelling random elements. The technique is easily executed, the results are readily interpreted and the computer removes tedium, lends accuracy and assists in impartiality.A scanning densitometer was adapted to allow computer control of the scan and to give direct computer storage of the data. A photographic transparency of the image to be scanned is mounted on a stage coupled directly to an accurate screw thread driven by a stepping motor. The stage is moved so that the fixed beam of the densitometer (which is directed normal to the transparency) traces a straight line along the structure of interest in the image.

Freeze fracture imaging and computer simulation of liposome structure: Membrane geometry and defects

1983 ◽  
Vol 41 ◽  
pp. 646-647
Author(s):  
Joseph A. Zasadzinski
Keyword(s):  
Liquid Crystalline ◽  
Freeze Fracture ◽  
Continuum Theory ◽  
Closed Surfaces ◽  
Straight Line ◽  
Low Weight ◽  
Concentric Spheres ◽  
Membrane Geometry ◽  
Dupin Cyclides ◽  
Limiting Case

At low weight fractions, many surfactant and biological amphiphiles form dispersions of lamellar liquid crystalline liposomes in water. Amphiphile molecules tend to align themselves in parallel bilayers which are free to bend. Bilayers must form closed surfaces to separate hydrophobic and hydrophilic domains completely. Continuum theory of liquid crystals requires that the constant spacing of bilayer surfaces be maintained except at singularities of no more than line extent. Maxwell demonstrated that only two types of closed surfaces can satisfy this constraint: concentric spheres and Dupin cyclides. Dupin cyclides (Figure 1) are parallel closed surfaces which have a conjugate ellipse (r1) and hyperbola (r2) as singularities in the bilayer spacing. Any straight line drawn from a point on the ellipse to a point on the hyperbola is normal to every surface it intersects (broken lines in Figure 1). A simple example, and limiting case, is a family of concentric tori (Figure 1b).To distinguish between the allowable arrangements, freeze fracture TEM micrographs of representative biological (L-α phosphotidylcholine: L-α PC) and surfactant (sodium heptylnonyl benzenesulfonate: SHBS)liposomes are compared to mathematically derived sections of Dupin cyclides and concentric spheres.

Human Enamel Replication

1976 ◽  
Vol 34 ◽  
pp. 180-181
Author(s):  
Norman L. Dockum ◽  
John G. Dockum
Keyword(s):  
Organic Matrix ◽  
Human Teeth ◽  
Human Enamel ◽  
Straight Line ◽  
Ultrastructural Characteristics

Ultrastructural characteristics of fractured human enamel and acid-etched enamel were compared using acetate replicas shadowed with platinum and palladium. Shadowed replications of acid-etched surfaces were also obtained by the same method.Enamel from human teeth has a rod structure within which there are crystals of hydroxyapatite contained within a structureless organic matrix composed of keratin. The rods which run at right angles from the dentino-enamel junction are considered to run in a straight line perpendicular to the perimeter of the enamel, however, in many areas these enamel rods overlap, interlacing and intertwining with one another.

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