scholarly journals Dot Product Representations of Planar Graphs

2011 ◽  
pp. 287-292 ◽  
Author(s):  
Ross J. Kang ◽  
Tobias Müller
Keyword(s):  
Planar Graphs ◽  
Dot Product

scholarly journals Dot Product Representations of Planar Graphs

10.37236/703 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Ross J. Kang ◽  
László Lovász ◽  
Tobias Müller ◽  
Edward R. Scheinerman
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Discrete Math ◽  
The Other ◽  
Outerplanar Graphs ◽  
Product Graph ◽  
Other Hand ◽  
Vector Labelling ◽  
Dot Product

A graph $G$ is a $k$-dot product graph if there exists a vector labelling $u: V(G) \to \mathbb{R}^k$ such that $u(i)^{T}u(j) \geq 1$ if and only if $ij \in E(G)$. Fiduccia, Scheinerman, Trenk and Zito [Discrete Math., 1998] asked whether every planar graph is a $3$-dot product graph. We show that the answer is "no". On the other hand, every planar graph is a $4$-dot product graph. We also answer the corresponding questions for planar graphs of prescribed girth and for outerplanar graphs.

Black-white contrast of dislocation loops

1978 ◽  
Vol 36 (1) ◽  
pp. 328-329
Author(s):  
J. J. Hren ◽  
W. D. Cooper ◽  
L. J. Sykes
Keyword(s):  
Electron Microscopy ◽  
Transmission Electron Microscopy ◽  
Dislocation Loops ◽  
Major Premise ◽  
Burgers Vector ◽  
Transmission Electron ◽  
Primary Means ◽  
Contrast Analysis ◽  
Dot Product ◽  
Imaging Conditions

Small dislocation loops observed by transmission electron microscopy exhibit a characteristic black-white strain contrast when observed under dynamical imaging conditions. In many cases, the topography and orientation of the image may be used to determine the nature of the loop crystallography. Two distinct but somewhat overlapping procedures have been developed for the contrast analysis and identification of small dislocation loops. One group of investigators has emphasized the use of the topography of the image as the principle tool for analysis. The major premise of this method is that the characteristic details of the image topography are dependent only on the magnitude of the dot product between the loop Burgers vector and the diffracting vector. This technique is commonly referred to as the (g•b) analysis. A second group of investigators has emphasized the use of the orientation of the direction of black-white contrast as the primary means of analysis.

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

A Space-Efficient Separator Algorithm for Planar Graphs

2019 ◽  
Vol E102.A (9) ◽  
pp. 1007-1016 ◽  
Author(s):  
Ryo ASHIDA ◽  
Sebastian KUHNERT ◽  
Osamu WATANABE
Keyword(s):  
Planar Graphs

Note on (semi-)proper orientation of some triangulated planar graphs

2021 ◽  
Vol 392 ◽  
pp. 125723
Author(s):  
Ruijuan Gu ◽  
Hui Lei ◽  
Yulai Ma ◽  
Zhenyu Taoqiu
Keyword(s):  
Planar Graphs ◽  
Proper Orientation

scholarly journals Distributed Dominating Set Approximations beyond Planar Graphs

2019 ◽  
Vol 15 (3) ◽  
pp. 1-18 ◽  
Author(s):  
Saeed Akhoondian Amiri ◽  
Stefan Schmid ◽  
Sebastian Siebertz
Keyword(s):  
Planar Graphs ◽  
Dominating Set

scholarly journals Clustered 3-colouring graphs of bounded degree

2021 ◽  
pp. 1-13
Author(s):  
Vida Dujmović ◽  
Louis Esperet ◽  
Pat Morin ◽  
Bartosz Walczak ◽  
David R. Wood
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Bounded Degree ◽  
Bounded Number ◽  
Proper Vertex ◽  
Vertex Colouring

Abstract A (not necessarily proper) vertex colouring of a graph has clustering c if every monochromatic component has at most c vertices. We prove that planar graphs with maximum degree $\Delta$ are 3-colourable with clustering $O(\Delta^2)$ . The previous best bound was $O(\Delta^{37})$ . This result for planar graphs generalises to graphs that can be drawn on a surface of bounded Euler genus with a bounded number of crossings per edge. We then prove that graphs with maximum degree $\Delta$ that exclude a fixed minor are 3-colourable with clustering $O(\Delta^5)$ . The best previous bound for this result was exponential in $\Delta$ .

Sign in / Sign up

Export Citation Format

Share Document