scholarly journals A Note on Obstructions to Clustered Planarity

10.37236/646 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Jamie Sneddon ◽  
Paul Bonnington
Keyword(s):  
Planar Embedding ◽  
Local Rotation ◽  
Clustered Planarity ◽  
Planar Digraph

A planar digraph $D$ is clustered planar if in some planar embedding of $D$ we have at each vertex the in-arcs occurring sequentially in the local rotation. By supplementing the operations used to form the usual minors in Kuratowski's theorem, clustered planar digraphs are characterised.

scholarly journals C-Planarity Testing of Embedded Clustered Graphs with Bounded Dual Carving-Width

Algorithmica ◽  
2021 ◽  
Author(s):  
Giordano Da Lozzo ◽  
David Eppstein ◽  
Michael T. Goodrich ◽  
Siddharth Gupta
Keyword(s):  
Dual Graph ◽  
Graph Visualization ◽  
Closed Disk ◽  
Running Time ◽  
Planar Embedding ◽  
Bounded Treewidth ◽  
Fpt Algorithm ◽  
Vertex Set ◽  
The One ◽  
Clustered Planarity

AbstractFor a clustered graph, i.e, a graph whose vertex set is recursively partitioned into clusters, the C-Planarity Testing problem asks whether it is possible to find a planar embedding of the graph and a representation of each cluster as a region homeomorphic to a closed disk such that (1) the subgraph induced by each cluster is drawn in the interior of the corresponding disk, (2) each edge intersects any disk at most once, and (3) the nesting between clusters is reflected by the representation, i.e., child clusters are properly contained in their parent cluster. The computational complexity of this problem, whose study has been central to the theory of graph visualization since its introduction in 1995 [Feng, Cohen, and Eades, Planarity for clustered graphs, ESA’95], has only been recently settled [Fulek and Tóth, Atomic Embeddability, Clustered Planarity, and Thickenability, to appear at SODA’20]. Before such a breakthrough, the complexity question was still unsolved even when the graph has a prescribed planar embedding, i.e, for embedded clustered graphs. We show that the C-Planarity Testing problem admits a single-exponential single-parameter FPT (resp., XP) algorithm for embedded flat (resp., non-flat) clustered graphs, when parameterized by the carving-width of the dual graph of the input. These are the first FPT and XP algorithms for this long-standing open problem with respect to a single notable graph-width parameter. Moreover, the polynomial dependency of our FPT algorithm is smaller than the one of the algorithm by Fulek and Tóth. In particular, our algorithm runs in quadratic time for flat instances of bounded treewidth and bounded face size. To further strengthen the relevance of this result, we show that an algorithm with running time O(r(n)) for flat instances whose underlying graph has pathwidth 1 would result in an algorithm with running time O(r(n)) for flat instances and with running time $$O(r(n^2) + n^2)$$ O ( r ( n 2 ) + n 2 ) for general, possibly non-flat, instances.

LFNet: Local Rotation Invariant Coordinates Frame for Robust Point Cloud Analysis

2021 ◽  
pp. 1-1
Author(s):  
Hezhi Cao ◽  
Ronghui Zhan ◽  
Yanxin Ma ◽  
Chao Ma ◽  
Jun Zhang
Keyword(s):  
Point Cloud ◽  
Rotation Invariant ◽  
Local Rotation ◽  
Point Cloud Analysis ◽  
Cloud Analysis

scholarly journals Clustered Planarity with Pipes

Algorithmica ◽  
2019 ◽  
Vol 81 (6) ◽  
pp. 2484-2526 ◽  
Author(s):  
Patrizio Angelini ◽  
Giordano Da Lozzo
Keyword(s):  
Clustered Planarity

scholarly journals Clustered Planarity: Clusters with Few Outgoing Edges

2009 ◽  
pp. 102-113 ◽  
Author(s):  
Vít Jelínek ◽  
Ondřej Suchý ◽  
Marek Tesař ◽  
Tomáš Vyskočil
Keyword(s):  
Clustered Planarity

scholarly journals Rotated infrared antenna transmitarray for the manipulation of circularly polarized wavefronts

2014 ◽  
Vol 1 ◽  
pp. 8 ◽  
Author(s):  
Yuchu He ◽  
George V. Eleftheriades
Keyword(s):  
Antenna Array ◽  
Polarized Light ◽  
Phase Control ◽  
Photonic Devices ◽  
Unit Cell ◽  
Fabrication Technology ◽  
Circularly Polarized Light ◽  
Circularly Polarized ◽  
Local Rotation ◽  
Infrared Frequencies

An IR optical transmitarray is presented based on Antenna Array Sheet (AAS) for the manipulation of circularly polarized light. The unit cell of the transmitarray comprises three layers of metallic elliptical patches. Complete phase control is achieved through the local rotation of each unit cell. Thin refraction and focusing transmitarrays of this sort are demonstrated at infrared frequencies. Moreover, a new concept for realizing a polarization-discriminating device is introduced based on a flat refracting lens. These devices are compatible with current fabrication technology and can become crucial for the integration with other IR and nano-photonic devices.

Vibronic polarons: Self-trapping, local rotation, and band features

2002 ◽  
Vol 89 (4) ◽  
pp. 405-418 ◽  
Author(s):  
A. G. Andreev ◽  
M. D. Ivanovich ◽  
M. Georgiev
Keyword(s):  
Local Rotation ◽  
Self Trapping
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