scholarly journals Straight-Line Drawing of Quadrangulations

2007 ◽  
pp. 234-239 ◽  
Author(s):  
Éric Fusy
Keyword(s):  
Line Drawing ◽  
Straight Line

scholarly journals THE POINT-SET EMBEDDABILITY PROBLEM FOR PLANE GRAPHS

2013 ◽  
Vol 23 (04n05) ◽  
pp. 357-395 ◽  
Author(s):  
THERESE BIEDL ◽  
MARTIN VATSHELLE
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Line Drawing ◽  
Np Hard ◽  
Bounded Treewidth ◽  
Straight Line ◽  
Triangulated Graphs ◽  
Point Set ◽  
Fixed Embedding ◽  
Set Of Points

In this paper, we study the point-set embeddability problem, i.e., given a planar graph and a set of points, is there a mapping of the vertices to the points such that the resulting straight-line drawing is planar? It was known that this problem is NP-hard if the embedding can be chosen, but becomes polynomial for triangulated graphs of treewidth 3. We show here that in fact it can be answered for all planar graphs with a fixed combinatorial embedding that have constant treewidth and constant face-degree. We prove that as soon as one of the conditions is dropped (i.e., either the treewidth is unbounded or some faces have large degrees), point-set embeddability with a fixed embedding becomes NP-hard. The NP-hardness holds even for a 3-connected planar graph with constant treewidth, triangulated planar graphs, or 2-connected outer-planar graphs. These results also show that the convex point-set embeddability problem (where faces must be convex) is NP-hard, but we prove that it becomes polynomial if the graph has bounded treewidth and bounded maximum degree.

scholarly journals Bounded-Degree Graphs have Arbitrarily Large Geometric Thickness

10.37236/1029 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
János Barát ◽  
Jiří Matoušek ◽  
David R. Wood
Keyword(s):  
Graph Drawing ◽  
Line Drawing ◽  
Geometric Graphs ◽  
Regular Graphs ◽  
Open Problems ◽  
Bounded Degree ◽  
Large N ◽  
Straight Line ◽  
Minimum Integer ◽  
Geometric Thickness

The geometric thickness of a graph $G$ is the minimum integer $k$ such that there is a straight line drawing of $G$ with its edge set partitioned into $k$ plane subgraphs. Eppstein [Separating thickness from geometric thickness. In Towards a Theory of Geometric Graphs, vol. 342 of Contemp. Math., AMS, 2004] asked whether every graph of bounded maximum degree has bounded geometric thickness. We answer this question in the negative, by proving that there exists $\Delta$-regular graphs with arbitrarily large geometric thickness. In particular, for all $\Delta\geq9$ and for all large $n$, there exists a $\Delta$-regular graph with geometric thickness at least $c\sqrt{\Delta}\,n^{1/2-4/\Delta-\epsilon}$. Analogous results concerning graph drawings with few edge slopes are also presented, thus solving open problems by Dujmović et al. [Really straight graph drawings. In Proc. 12th International Symp. on Graph Drawing (GD '04), vol. 3383 of Lecture Notes in Comput. Sci., Springer, 2004] and Ambrus et al. [The slope parameter of graphs. Tech. Rep. MAT-2005-07, Department of Mathematics, Technical University of Denmark, 2005].

scholarly journals Curved vs. Straight-Line Handwriting Effects on Word Recognition in Typical and Dyslexic Readers Across Chinese and English

2021 ◽  
Vol 12 ◽  
Author(s):  
Connie Qun Guan ◽  
Yifei Li ◽  
Wanjin Meng ◽  
Laura M. Morett
Keyword(s):  
Word Recognition ◽  
Multisensory Integration ◽  
Character Recognition ◽  
Word Processing ◽  
Line Drawing ◽  
English Word ◽  
Visual Word ◽  
Word Form ◽  
Straight Line ◽  
Study Results

Handwriting serves to link auditory and motor routines with visual word processing, which is a hallmark of successful reading. The current study aims to explore the effect of multisensory integration as a pathway to neural specialization for print among typical and dyslexic readers across writing systems. We identified 9–10-year-old dyslexic Chinese children (n = 24) and their typically developing counterparts (n = 24) on whom we conducted both behavioral and electroencephalogram (EEG) experiments. We designed four learning conditions: Handwriting Chinese (HC), Viewing Chinese (VC), Drawing followed by Character Recognition in Chinese (D-C), and Drawing followed by Word Recognition in English (D-E). In both handwriting and drawing conditions, we also designed curved vs. straight-line stimuli. Both behavioral and EEG results showed that handwriting straight line strokes facilitated visual word recognition in Chinese compared to handwriting curved lines. Handwriting conditions resulted in a lateralization of the N170 in typical readers, but not the dyslexic readers. Interestingly, drawing curved lines facilitate word recognition in English among dyslexic readers. Taken together, the results of the study suggest benefits of handwriting on the neural processing and behavioral performance in response to Chinese character recognition and curved-line drawing effects on English word recognition among dyslexic readers. But the lack of handwriting effects in dyslexic readers suggest that students who have deficits in reading may also be missing the link between multisensory integration and word recognition in the visual word form areas. The current study results have implications for maintaining handwriting practices to promote perception and motor integration for visual word form area development for normal readers and suggest that drawing practices might benefit Chinese dyslexic readers in reading English.

Straight-line monotone grid drawings of series–parallel graphs

2015 ◽  
Vol 07 (02) ◽  
pp. 1550007 ◽  
Author(s):  
Md. Iqbal Hossain ◽  
Md. Saidur Rahman
Keyword(s):  
Planar Graph ◽  
Linear Time ◽  
Line Drawing ◽  
Monotone Path ◽  
Straight Line ◽  
Parallel Graph

A monotone drawing of a planar graph G is a planar straight-line drawing of G where a monotone path exists between every pair of vertices of G in some direction. Recently monotone drawings of graphs have been discovered as a new standard for visualizing graphs. In this paper we study monotone drawings of series–parallel graphs in a variable embedding setting. We show that a series–parallel graph of n vertices has a straight-line planar monotone drawing on a grid of size O(n) × O(n2) and such a drawing can be found in linear time.

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