A simple proof of the representation of bipartite planar graphs as the contact graphs of orthogonal straight line segments

1998 ◽  
Vol 66 (3) ◽  
pp. 125-126 ◽  
Author(s):  
Jurek Czyzowicz ◽  
Evangelos Kranakis ◽  
Jorge Urrutia
Keyword(s):  
Simple Proof ◽  
Planar Graphs ◽  
Line Segments ◽  
Straight Line ◽  
Contact Graphs

scholarly journals Convex Grid Drawings of 3-Connected Planar Graphs

1997 ◽  
Vol 07 (03) ◽  
pp. 211-223 ◽  
Author(s):  
Marek Chrobak ◽  
Goos Kant
Keyword(s):  
Planar Graphs ◽  
Linear Time ◽  
Time Algorithm ◽  
Plane Graph ◽  
Linear Time Algorithm ◽  
Convex Polygons ◽  
Line Segments ◽  
Polynomial Size ◽  
Straight Line

We consider the problem of embedding the vertices of a plane graph into a small (polynomial size) grid in the plane in such a way that the edges are straight, nonintersecting line segments and faces are convex polygons. We present a linear-time algorithm which, given an n-vertex 3-connected plane G (with n ≥ 3), finds such a straight-line convex embedding of G into a (n - 2) × (n - 2) grid.

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.

SIMULTANEOUS EMBEDDING OF EMBEDDED PLANAR GRAPHS

2013 ◽  
Vol 23 (02) ◽  
pp. 93-126 ◽  
Author(s):  
PATRIZIO ANGELINI ◽  
GIUSEPPE DI BATTISTA ◽  
FABRIZIO FRATI
Keyword(s):  
Planar Graphs ◽  
Np Hard ◽  
Line Segments ◽  
Planar Embedding ◽  
Straight Line ◽  
K 13 ◽  
Hard Problems ◽  
Simultaneous Embedding ◽  
Np Hardness ◽  
Np Hard Problems

A simultaneous embedding with fixed edges (SEFE) of a set of k planar graphs G1,…,Gk on the same set of vertices is a set of k planar drawings of G1,…,Gk, respectively, such that each vertex is placed on the same point in all the drawings and each edge is represented by the same Jordan curve in the drawings of all the graphs it belongs to. A simultaneous geometric embedding (SGE) is a SEFE in which the edges are represented by straight-line segments. Given k planar graphs G1,…,Gk, deciding whether they admit a SEFE and whether they admit an SGE are NP-hard problems, for k ≥ 3 and for k ≥ 2, respectively. In this paper we consider the complexity of SEFE and of SGE when the graphs G1,…,Gk have a fixed planar embedding. In sharp contrast with the NP-hardness of SEFE for three non-embedded planar graphs, we show that SEFE is quadratic-time solvable for three graphs with a fixed planar embedding. Furthermore, we show that, given k embedded planar graphs G1,…,Gk, deciding whether a SEFE of G1,…,Gk exists and deciding whether an SGE of G1,…,Gk exists are NP-hard problems, for k ≥ 14 and k ≥ 13, respectively.

Acute Constraints in Straight-Line Drawings of Planar Graphs

2019 ◽  
Vol E102.A (9) ◽  
pp. 994-1001
Author(s):  
Akane SETO ◽  
Aleksandar SHURBEVSKI ◽  
Hiroshi NAGAMOCHI ◽  
Peter EADES
Keyword(s):  
Planar Graphs ◽  
Line Drawings ◽  
Straight Line

scholarly journals Use of otoliths to estimate size at sexual maturity in fish

2000 ◽  
Vol 43 (4) ◽  
pp. 437-440 ◽  
Author(s):  
Carlos Sérgio Agostinho
Keyword(s):  
Sexual Maturity ◽  
Line Segments ◽  
Alternative Method ◽  
Straight Line ◽  
Size At Sexual Maturity

The viability of an alternative method for estimating the size at sexual maturity of females of Plagioscion squamosissimus (Perciformes, Sciaenidae) was analyzed. This methodology was used to evaluate the size at sexual maturity in crabs, but has not yet been used for this purpose in fishes. Separation of young and adult fishes by this method is accomplished by iterative adjustment of straight-line segments to the data for length of the otolith and length of the fish. The agreement with the estimate previously obtained by another technique and the possibility of calculating the variance indicates that in some cases, the method analyzed can be used successfully to estimate size at sexual maturity in fish. However, additional studies are necessary to detect possible biases in the method.

scholarly journals An Automatic Measurement Method for Absolute Depth of Objects in Two Monocular Images Based on SIFT Feature

2017 ◽  
Author(s):  
Lixin He ◽  
Jing Yang ◽  
Bin Kong ◽  
Can Wang
Keyword(s):  
Automatic Measurement ◽  
Depth Estimation ◽  
Depth Information ◽  
Scale Invariant ◽  
Line Segments ◽  
Sift Feature ◽  
Novel Approach ◽  
Straight Line ◽  
Object Depth ◽  
Scale Invariant Feature

It is one of very important and basic problem in compute vision field that recovering depth information of objects from two-dimensional images. In view of the shortcomings of existing methods of depth estimation, a novel approach based on SIFT (the Scale Invariant Feature Transform) is presented in this paper. The approach can estimate the depths of objects in two images which are captured by an un-calibrated ordinary monocular camera. In this approach, above all, the first image is captured. All of the camera parameters remain unchanged, and the second image is acquired after moving the camera a distance d along the optical axis. Then image segmentation and SIFT feature extraction are implemented on the two images separately, and objects in the images are matched. Lastly, an object depth can be computed by the lengths of a pair of straight line segments. In order to ensure that the best appropriate a pair of straight line segments are chose and reduce the computation, the theory of convex hull and the knowledge of triangle similarity are employed. The experimental results show our approach is effective and practical.

scholarly journals On a Class of Planar Graphs with Straight-Line Grid Drawings on Linear Area

2009 ◽  
Vol 13 (2) ◽  
pp. 153-177 ◽  
Author(s):  
Md. Rezaul Karim ◽  
Md. Saidur Rahman
Keyword(s):  
Planar Graphs ◽  
Straight Line

scholarly journals Low-level Grouping of Straight Line Segments

1991 ◽  
Author(s):  
A. Etemadi ◽  
J. P. Schmidt ◽  
G. Matas ◽  
J. Illingworth ◽  
J. Kittler
Keyword(s):  
Line Segments ◽  
Low Level ◽  
Straight Line
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