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### Simultaneous Drawing of Planar Graphs with Right-Angle Crossings and Few Bends

Michael A. Bekos, Thomas C. van Dijk, Philipp Kindermann, Alexander Wolff
2016 Journal of Graph Algorithms and Applications
graphs form right angles.  ...  Given two planar graphs that are defined on the same set of vertices, a RAC simultaneous drawing is one in which each graph is drawn planar, there are no edge overlaps and the crossings between the two  ...  JGAA'13]A Few Known ResultsAny pair of trees admits an SE drawing with 1 bend and quadratic area.Any pair of planar graphs admits an SE drawing with 2 bends and quadratic area. small angles A cycle and  ...

### Simultaneous Drawing of Planar Graphs with Right-Angle Crossings and Few Bends [chapter]

Michael A. Bekos, Thomas C. van Dijk, Philipp Kindermann, Alexander Wolff
2015 Lecture Notes in Computer Science
JGAA'13]A Few Known ResultsAny pair of trees admits an SE drawing with 1 bend and quadratic area.Any pair of planar graphs admits an SE drawing with 2 bends and quadratic area. small angles A cycle and  ...  Any pair of trees admits an SE drawing with 1 bend and quadratic area.Any pair of planar graphs admits an SE drawing with 2 bends and quadratic area.small angles A cycle and a matching always admit a geometric  ...  Wheel × Matching All graphs are drawn on the O(n) × O(n)-grid. All algorithms run in O(n) time. Sefe Rac-  ...

### Simultaneous Embeddings with Few Bends and Crossings

Fabrizio Frati, Michael Hoffmann, Vincent Kusters
2019 Journal of Graph Algorithms and Applications
A simultaneous embedding with fixed edges (Sefe) of two planar graphs R and B is a pair of plane drawings of R and B that coincide when restricted to their common vertices and edges.  ...  We show that whenever R and B admit a Sefe, they also admit a Sefe in which every edge is a polygonal curve with few bends and every pair of edges has few crossings.  ...  This research initiated at the Workshop on Geometry and Graphs, held at the Bellairs Research Institute in Barbados in March 2015.  ...

### Simultaneous Embeddings with Few Bends and Crossings [article]

Fabrizio Frati, Michael Hoffmann, Vincent Kusters
2015 arXiv   pre-print
A simultaneous embedding with fixed edges (SEFE) of two planar graphs R and B is a pair of plane drawings of R and B that coincide when restricted to the common vertices and edges of R and B.  ...  We show that whenever R and B admit a SEFE, they also admit a SEFE in which every edge is a polygonal curve with few bends and every pair of edges has few crossings.  ...  Acknowledgments This research initiated at the Workshop on Geometry and Graphs, held at the Bellairs Research Institute in Barbados in March 2015.  ...

### 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.  ...  (c) A straight-line drawing computed with a force-directed algorithm (left) and a subsequent drawing with improved crossing angle resolution, obtained using smoothed curves (right).  ...

### Simultaneous graph embedding with bends and circular arcs

Justin Cappos, Alejandro Estrella-Balderrama, J. Joseph Fowler, Stephen G. Kobourov
2009 Computational geometry
We demonstrate how to simultaneously embed a path and an n-level planar graph and how to use radial embeddings for curvilinear simultaneous embeddings of a path and an outerplanar graph.  ...  We also show how to use star-shaped levels to find 2-bends per path edge simultaneous embeddings of a path and an outerplanar graph. All embedding algorithms run in O (n) time.  ...  With level and radial embeddings of graphs, we improve on this result to obtain a planar simultaneous embedding of an n-vertex planar graph G with no edge bends and an n-vertex path P , which may have  ...

### Planar Polyline Drawings with Good Angular Resolution [chapter]

Carsten Gutwenger, Petra Mutzel
1998 Lecture Notes in Computer Science
To our best knowledge, this algorithm achieves the best simultaneous bounds concerning the grid size, angular resolution, and number of bends for planar grid drawings of high-degree planar graphs.  ...  We present a linear time algorithm that constructs a planar polyline grid drawing of any plane graph with n vertices and maximum degree d on a (2n − 5) × ( 3 2 n − 7 2 ) grid with at most 5n − 15 bends  ...  Important aesthetic criteria for nice drawings are: a small number of edge crossings, small area, few bends, and a good angular resolution (see, e.g., [27] ).  ...

### Semi-bipartite Graph Visualization for Gene Ontology Networks [chapter]

Kai Xu, Rohan Williams, Seok-Hee Hong, Qing Liu, Ji Zhang
2010 Lecture Notes in Computer Science
Angle Crossing (RAC) drawings are poly-line drawings where each crossing forms four right angles.  ...  , and study tradeoffs between the crossing resolution (i.e., the minimum angle formed by two crossing segments), the curve complexity (i.e., maximum number of bends per edge), the total number of bends  ...

### Non-planar Orthogonal Drawings with Fixed Topology [chapter]

Markus Chimani, Gunnar W. Klau, René Weiskircher
2005 Lecture Notes in Computer Science
We present a procedure for calculating the bend minimal shape of non-planar graphs with given topology.  ...  In this paper we extend the drawing standard to the class of non-planar graphs and present an algorithm that computes provably bend-minimal drawings within this standard.  ...  They have between 10 and 100 nodes each, and most of them are non-planar. But even the non-planar graphs are quite sparse and have therefore very few dummy-nodes after the planarization step.  ...

### Drawing Partially Embedded and Simultaneously Planar Graphs

Timothy M. Chan, Fabrizio Frati, Carsten Gutwenger, Anna Lubiw, Petra Mutzel, Marcus Schaefer
2015 Journal of Graph Algorithms and Applications
We investigate the problem of constructing planar drawings with few bends for two related problems, the partially embedded graph problem-to extend a straight-line planar drawing of a subgraph to a planar  ...  drawing of the whole graph-and the simultaneous planarity problem-to find planar drawings of two graphs that coincide on shared vertices and edges.  ...  The authors would like to thank the anonymous referees for their useful comments and suggestions.  ...

### Drawing Partially Embedded and Simultaneously Planar Graphs [chapter]

Timothy M. Chan, Fabrizio Frati, Carsten Gutwenger, Anna Lubiw, Petra Mutzel, Marcus Schaefer
2014 Lecture Notes in Computer Science
We investigate the problem of constructing planar drawings with few bends for two related problems, the partially embedded graph problem-to extend a straight-line planar drawing of a subgraph to a planar  ...  drawing of the whole graph-and the simultaneous planarity problem-to find planar drawings of two graphs that coincide on shared vertices and edges.  ...  The authors would like to thank the anonymous referees for their useful comments and suggestions.  ...

### Beyond-Planar Graphs: Algorithmics and Combinatorics (Dagstuhl Seminar 16452)

Sok-Hee Hong, Michael Kaufmann, Stephen G. Kobourov, János Pach, Marc Herbstritt
2017 Dagstuhl Reports
types of crossings, or with some forbidden crossing patterns.  ...  The common interest was in the exploration of structural properties and the development of algorithms for so-called beyond-planar graphs, i.e., non-planar graphs with topological constraints such as specific  ...  RAC (Right Angle Crossing) graphs: a graph which has a straight-line drawing with right angle crossings.  ...

### Drawing Subcubic 1-Planar Graphs with Few Bends, Few Slopes, and Large Angles [article]

Philipp Kindermann, Fabrizio Montecchiani, Lena Schlipf, André Schulz
2018 arXiv   pre-print
We also show that 2 slopes always suffice for 1-planar drawings of subcubic 1-planar graphs with at most 2 bends per edge.  ...  We show that the 1-planar slope number of 3-connected cubic 1-planar graphs is at most 4 when edges are drawn as polygonal curves with at most 1 bend each.  ...  (3) Is the 1-planar slope number of 1-planar graphs bounded by a function of the maximum degree?  ...

### Embedding Vertices at Points: Few Bends Suffice for Planar Graphs

Michael Kaufmann, Roland Wiese
2002 Journal of Graph Algorithms and Applications
We relax the latter requirement and allow very few bends on each edge while considering general plane graphs.  ...  Our results show two algorithms for mapping four-connected plane graphs with at most one bend per edge and for mapping general plane graphs with at most two bends per edge.  ...  Next we generalize the technique so that it will work for any planar graph and produces drawings with at most two bends per edge.  ...

### Embedding Vertices at Points: Few Bends Suffice for Planar Graphs [chapter]

Michael Kaufmann, Roland Wiese
2004 Graph Algorithms and Applications 3
We relax the latter requirement and allow very few bends on each edge while considering general plane graphs.  ...  Our results show two algorithms for mapping four-connected plane graphs with at most one bend per edge and for mapping general plane graphs with at most two bends per edge.  ...  Next we generalize the technique so that it will work for any planar graph and produces drawings with at most two bends per edge.  ...
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