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On RAC Drawings of Graphs with one Bend per Edge [article]

Patrizio Angelini, Michael A. Bekos, Henry Förster, Michael Kaufmann
2018 arXiv   pre-print
A k-bend right-angle-crossing drawing or (k-bend RAC drawing, for short) of a graph is a polyline drawing where each edge has at most k bends and the angles formed at the crossing points of the edges are  ...  We also demonstrate that there exist infinitely many n-vertex 1-bend RAC graphs with exactly 5n-O(1) edges.  ...  that admit polyline drawings, with few bends per edge, in which the angles formed at the edge crossings are 90 • .  ... 
arXiv:1808.10470v1 fatcat:vmoqfxp2ubbv3gd5fhv6nomura

Drawing graphs with right angle crossings

Walter Didimo, Peter Eades, Giuseppe Liotta
2011 Theoretical Computer Science  
This paper initiates the study of combinatorial and algorithmic questions related to the problem of computing RAC drawings with few bends per edge.  ...  Namely, we study the interplay between number of bends per edge and total number of edges in RAC drawings.  ...  (c) If G admits a RAC drawing with at most one bend per edge then m = O(n 4 3 ). (d) If G admits a RAC drawing with at most two bends per edge then m = O(n 7 4 ).  ... 
doi:10.1016/j.tcs.2011.05.025 fatcat:aapalbfrorfmvcdiacvhm4ss64

1-Bend RAC Drawings of 1-Planar Graphs [article]

Walter Didimo, Giuseppe Liotta, Saeed Mehrabi, Fabrizio Montecchiani
2016 arXiv   pre-print
One of the main questions still open is whether every 1-planar graph has a RAC drawing with at most one bend per edge. We positively answer this question.  ...  Also, straight-line RAC drawings always exist for IC-planar graphs, a subclass of 1-planar graphs.  ...  Let G be an n-vertex 1-planar graph. Then G admits a 1-planar RAC drawing Γ with at most one bend per edge.  ... 
arXiv:1608.08418v1 fatcat:rzcne3zy2zbk3kbscfapxjdew4

Drawing Graphs with Right Angle Crossings [chapter]

Walter Didimo, Peter Eades, Giuseppe Liotta
2009 Lecture Notes in Computer Science  
This paper initiates the study of combinatorial and algorithmic questions related with the problem of computing RAC drawings with few bends per edge.  ...  Namely, we study the interplay between number of bends per edge and total number of edges in RAC drawings.  ...  Drawing Graphs with Right Angle Crossings  ... 
doi:10.1007/978-3-642-03367-4_19 fatcat:jlucpnj6knf5nmjdrq6rryf3wi

A Survey on Graph Drawing Beyond Planarity [article]

Walter Didimo, Giuseppe Liotta, Fabrizio Montecchiani
2018 arXiv   pre-print
Graph Drawing Beyond Planarity is a rapidly growing research area that classifies and studies geometric representations of non-planar graphs in terms of forbidden crossing configurations.  ...  Aim of this survey is to describe the main research directions in this area, the most prominent known results, and some of the most challenging open problems.  ...  Figure 6 (c) shows a 1-planar RAC drawing with at most one bend per edge of the graph in Fig. 6(b) .  ... 
arXiv:1804.07257v1 fatcat:xeckxilhfbbgdbfhctikymlq6m

RAC Drawings in Subcubic Area [article]

Zahed Rahmati, Fatemeh Emami
2020 arXiv   pre-print
., on the area of RAC drawings.  ...  Given a graph with n vertices and m edges, we provide a RAC drawing algorithm with curve complexity 6 and area O(n^2.75), which takes time O(n+m).  ...  in [11] ; there exists no RAC drawing of K n with 3 bends per edge in O(n 2 ) area for sufficiently large n; every graph admits a RAC drawing with 8 bends per edge in O(n 2 ) area.  ... 
arXiv:2001.02867v1 fatcat:7bwr5cqegbdmjjfjomyucj3bpy

Drawing Graphs with Vertices at Specified Positions and Crossings at Large Angles [chapter]

Martin Fink, Jan-Henrik Haunert, Tamara Mchedlidze, Joachim Spoerhase, Alexander Wolff
2012 Lecture Notes in Computer Science  
We investigate two drawing styles where only bends or both bends and edges must be drawn on an underlying grid. We present various results for drawings with one, two, and three bends per edge.  ...  Part of this research has been presented as a poster at the 19th Int. Symposium on Graph Drawing, Eindhoven, 2011.  ...  If such a drawing exists and the largest number of bends per edge in the drawing is b, we say that G admits a RAC b (or an αAC b ) embedding on S.  ... 
doi:10.1007/978-3-642-28076-4_19 fatcat:bfn6x7zlwna5tdbfhtomkuvr74

Graphs that admit right angle crossing drawings

Karin Arikushi, Radoslav Fulek, Balázs Keszegh, Filip Morić, Csaba D. Tóth
2012 Computational geometry  
We show that if a graph with n vertices admits a RAC drawing with at most 1 bend or 2 bends per edge, then the number of edges is at most 6.5n and 74.2n, respectively.  ...  We consider right angle crossing (RAC) drawings of graphs in which the edges are represented by polygonal arcs and any two edges can cross only at a right angle.  ...  Acknowledgement We would like to acknowledge an anonymous referee for pointing out that K 6 has a RAC 1 drawing.  ... 
doi:10.1016/j.comgeo.2011.11.008 fatcat:6fbspb7qcja2dpdokkoqhtlewe

Graphs that Admit Right Angle Crossing Drawings [chapter]

Karin Arikushi, Radoslav Fulek, Baláazs Keszegh, Filip Morić, Csaba D. Tóth
2010 Lecture Notes in Computer Science  
We show that if a graph with n vertices admits a RAC drawing with at most 1 bend or 2 bends per edge, then the number of edges is at most 6.5n and 74.2n, respectively.  ...  We consider right angle crossing (RAC) drawings of graphs in which the edges are represented by polygonal arcs and any two edges can cross only at a right angle.  ...  The size of a face is the number of edges (counted with multiplicity) on the boundary of f . RAC Drawings with One Bend per Edge Discharging.  ... 
doi:10.1007/978-3-642-16926-7_14 fatcat:xpkrxoks6renjfqyja4dhziwsq

Drawing Graphs with Vertices at Specified Positions and Crossings at Large Angles [article]

Martin Fink, Jan-Henrik Haunert, Tamara Mchedlidze, Joachim Spoerhase, Alexander Wolff
2011 arXiv   pre-print
We investigate two drawing styles where only bends or both bends and edges must be drawn on an underlying grid. We present various results for drawings with one, two, and three bends per edge.  ...  Point-set embeddings and large-angle crossings are two areas of graph drawing that independently have received a lot of attention in the past few years.  ...  If such a drawing exists and the largest number of bends per edge in the drawing is b, we say that G admits a RAC b (or an αAC b ) embedding on S.  ... 
arXiv:1107.4970v1 fatcat:i5ycjqcbojddtjvsgfunw45z4u

Drawing Graphs with Vertices at Specified Positions and Crossings at Large Angles [chapter]

Martin Fink, Jan-Henrik Haunert, Tamara Mchedlidze, Joachim Spoerhase, Alexander Wolff
2012 Lecture Notes in Computer Science  
Recent experiments on the readability of drawings [4] showed that polyline drawings with angles at edge crossings close to 90 • and a small number of bends per edge are just as readable as planar drawings  ...  If such a drawing exists and the largest number of bends per edge in the drawing is b, we say that G admits a RAC b (or an αAC b ) embedding on S.  ... 
doi:10.1007/978-3-642-25878-7_43 fatcat:w3mzoh6oprfhxfzbus4wrmiy34

On Compact RAC Drawings

Henry Förster, Michael Kaufmann, Peter Sanders, Grzegorz Herman, Fabrizio Grandoni
2020 European Symposium on Algorithms  
[Emilio Di Giacomo et al., 2011] by showing how to compute a RAC drawing with three bends per edge in cubic area.  ...  As a counterpart, we prove that in general quadratic area is not sufficient for RAC drawings with three bends per edge.  ...  Let Γ be a RAC drawing of K n with O(1) bends per edge.  ... 
doi:10.4230/lipics.esa.2020.53 dblp:conf/esa/Forster020 fatcat:r2vjw25zkfcoxi7t7wnwnnayje

Compact Drawings of 1-Planar Graphs with Right-Angle Crossings and Few Bends [article]

Steven Chaplick, Fabian Lipp, Alexander Wolff, Johannes Zink
2018 arXiv   pre-print
First, we show that every n-vertex NIC-planar graph admits a NIC-planar RAC drawing with at most one bend per edge on a grid of size O(n) × O(n).  ...  Then, we show that every n-vertex 1-planar graph admits a 1-planar RAC drawing with at most two bends per edge on a grid of size O(n^3) × O(n^3).  ...  Does any 1-planar graph admit a RAC poly 1 drawing? Can we draw any graph in RAC 0 with only right-angle crossings in polynomial area when we allow one or two bends per edge?  ... 
arXiv:1806.10044v4 fatcat:k2ysur7abfevdnkplhpxziguga

On the Perspectives Opened by Right Angle Crossing Drawings [chapter]

Patrizio Angelini, Luca Cittadini, Giuseppe Di Battista, Walter Didimo, Fabrizio Frati, Michael Kaufmann, Antonios Symvonis
2010 Lecture Notes in Computer Science  
Second, we study if RAC drawings allow us to draw bounded-degree graphs with lower curve complexity than the one required by more constrained drawing conventions.  ...  RAC drawings have been introduced because cognitive experiments provided evidence that increasing the number of crossings does not decrease the readability of the drawing if the edges cross at right angles  ...  Further, every graph admits a RAC drawing with at most three bends per edge, and this curve complexity is required in infinitely many cases.  ... 
doi:10.1007/978-3-642-11805-0_5 fatcat:sdkmwoxbkjgm7kcm2sxqlvntuq

On the Perspectives Opened by Right Angle Crossing Drawings

Patrizio Angelini, Luca Cittadini, Walter Didimo, Fabrizio Frati, Giuseppe Di Battista, Michael Kaufmann, Antonios Symvonis
2011 Journal of Graph Algorithms and Applications  
Second, we study if RAC drawings allow us to draw bounded-degree graphs with lower curve complexity than the one required by more constrained drawing conventions.  ...  RAC drawings have been introduced because cognitive experiments provided evidence that increasing the number of crossings does not decrease the readability of the drawing if the edges cross at right angles  ...  Further, every graph admits a RAC drawing with at most three bends per edge, and this curve complexity is required in infinitely many cases.  ... 
doi:10.7155/jgaa.00217 fatcat:hhlcmlbxdfhl7fj2t5i3d4d3he
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