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Level-Planar Drawings with Few Slopes [article]

Guido Brückner, Nadine Davina Krisam, Tamara Mchedlidze
2019 arXiv   pre-print
We introduce and study level-planar straight-line drawings with a fixed number λ of slopes.  ...  We complement these positive results by showing that testing whether non-proper level graphs admit level-planar drawings with λ slopes is NP-hard even in restricted cases.  ...  Level-planar drawing with few slopes have not been considered in the literature but drawings of undirected graphs with few slopes have been extensively studied.  ... 
arXiv:1907.13558v1 fatcat:a6mtdmeqgfdbvlmvgjn4nm323y

Level-Planar Drawings with Few Slopes

Guido Brückner, Nadine Krisam, Tamara Mchedlidze
2021
We introduce and study level-planar straight-line drawings with a fixed number 𝜆 of slopes.  ...  We complement these positive results by showing that testing whether non-proper level graphs admit level-planar drawings with 𝜆 slopes is NP-hard even in restricted cases.  ...  Level-planar drawing with few slopes have not been considered in the literature but drawings of undirected graphs with few slopes have been extensively studied.  ... 
doi:10.5445/ir/1000140541 fatcat:p6blstwpbvc7ldkom75x2lyu7a

Really Straight Graph Drawings [chapter]

Vida Dujmović, Matthew Suderman, David R. Wood
2005 Lecture Notes in Computer Science  
Drawings of non-planar graphs with few slopes are also considered. For example, it is proved that graphs of bounded degree and bounded treewidth have drawings with O(log n) slopes.  ...  We study straight-line drawings of graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2-trees, and planar 3-trees.  ...  Then G has a drawing with ks (k − 1) + k + s slopes. Lemma 2 suggests looking at host graphs that admit drawings with few slopes and few edge lengths.  ... 
doi:10.1007/978-3-540-31843-9_14 fatcat:fwradttc6ba7jip6ftrdahsdwq

Speed of comprehension of visualized ordered sets

Christof Körner, Dietrich Albert
2002 Journal of experimental psychology. Applied  
The authors investigated the effects of visual properties of hierarchical graphs on speed of comprehension: planarity (crossing of lines), slopes (orientation of lines), and levels (adjustment of dots)  ...  Preliminary conclusions for the presentation of hierarchical graphs are drawn. 1993): Visual planarity refers to whether lines in the drawing are  ...  We used the same eight drawings as in Experiments 1a, 1b, and 1c that varied with respect to visual planarity, slopes, and levels.  ... 
doi:10.1037/1076-898x.8.1.57 pmid:12009177 fatcat:iztuvpafajdo5hgvrsrucuygga

Speed of comprehension of visualized ordered sets

Christof Körner, Dietrich Albert
2002 Journal of experimental psychology. Applied  
The authors investigated the effects of visual properties of hierarchical graphs on speed of comprehension: planarity (crossing of lines), slopes (orientation of lines), and levels (adjustment of dots)  ...  Preliminary conclusions for the presentation of hierarchical graphs are drawn. 1993): Visual planarity refers to whether lines in the drawing are  ...  We used the same eight drawings as in Experiments 1a, 1b, and 1c that varied with respect to visual planarity, slopes, and levels.  ... 
doi:10.1037//1076-898x.8.1.57 pmid:12009177 fatcat:b65pm7t5ereuzetsio4hrggwnu

Complexity of Finding Non-Planar Rectilinear Drawings of Graphs [chapter]

Ján Maňuch, Murray Patterson, Sheung-Hung Poon, Chris Thachuk
2011 Lecture Notes in Computer Science  
Dot Product Representations of Planar Graphs p. 287 Drawing Planar Graphs of Bounded Degree with Few Slopes p. 293 Complexity of Finding Non-planar Rectilinear Drawings of Graphs p. 305 Point-Set  ...  K-Level Planarization and Crossing Minimization p. 238 On Touching Triangle Graphs p. 250 Triangle Contact Representations and Duality p. 262 On Maximum Differential Graph Coloring p. 274  ... 
doi:10.1007/978-3-642-18469-7_28 fatcat:pxln4k6bzbhxrlde4iufrtt74u

Drawing Planar Graphs with Few Geometric Primitives

Gregor Hültenschmidt, Philipp Kindermann, Wouter Meulemans, André Schulz
2018 Journal of Graph Algorithms and Applications  
Further, we present an algorithm for drawing planar 3-trees with (8n-17)/3 segments on an O(n)× O(n^2) grid.  ...  We also study the problem of drawing maximal planar graphs with circular arcs and provide an algorithm to draw such graphs using only (5n - 11)/3 arcs.  ...  Planar 3-trees with few segments on the grid In this section, we show how to construct drawings of planar 3-trees with few segments on the grid. We begin by introducing some notation.  ... 
doi:10.7155/jgaa.00473 fatcat:yfl4babp25bsrps5vzr6zd4ggm

Upward planar drawings with two slopes [article]

Jonathan Klawitter, Tamara Mchedlidze
2021 arXiv   pre-print
As an application of this drawing style, we show how to draw an upward planar phylogenetic network with two slopes such that all leaves lie on a horizontal line.  ...  In an upward planar 2-slope drawing of a digraph, edges are drawn as straight-line segments in the upward direction without crossings using only two different slopes.  ...  [BKM19] considered level-planar drawings with a fixed slope set, that is, upward planar drawings where each vertex is drawn on a predefined integer y-coordinate (its level).  ... 
arXiv:2106.02839v2 fatcat:t62riog4n5hlbknlbe3v3h46uy

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

Sok-Hee Hong, Michael Kaufmann, Stephen G. Kobourov, János Pach, Marc Herbstritt
2017 Dagstuhl Reports  
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  ...  types of crossings, or with some forbidden crossing patterns.  ...  Graph drawings with few slopes. Comput. Geom. Theory Appl., 38(3):181-193, 2007. 12 Stephane Durocher and Debajyoti Mondal. Drawing plane triangulations with few segments. In Proc. Canad. Conf.  ... 
doi:10.4230/dagrep.6.11.35 dblp:journals/dagstuhl-reports/Hong0KP16 fatcat:gmft5jywxbapza52lhj4o23ngq

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  
We study the planar slope number, i.e., the minimum number of distinct edge-slopes in such a drawing of a planar graph with maximum degree ∆.  ...  In this paper, we present a new algorithm for drawing UPRs that greatly improves the layout quality, leading to good hierarchal drawings with few crossings.  ... 
doi:10.1007/978-3-642-11805-0_24 fatcat:724qxxufdzgthfzrovmnj4nzky

Drawing Planar Graphs with Few Geometric Primitives [chapter]

Gregor Hültenschmidt, Philipp Kindermann, Wouter Meulemans, André Schulz
2017 Lecture Notes in Computer Science  
Further, we present an algorithm for drawing planar 3-trees with (8n−17)/3 segments on an O(n) × O(n 2 ) grid.  ...  We also study the problem of drawing maximal planar graphs with circular arcs and provide an algorithm to draw such graphs using only (5n − 11)/3 arcs.  ...  Planar 3-trees with few segments on the grid In this section, we show how to construct drawings of planar 3-trees with few segments on the grid. We begin by introducing some notation.  ... 
doi:10.1007/978-3-319-68705-6_24 fatcat:mtblxoakmzbypcpbh3afwco7d4

Simultaneous Embedding of Planar Graphs with Few Bends [chapter]

Cesim Erten, Stephen G. Kobourov
2005 Lecture Notes in Computer Science  
We present an O(n) time algorithm for simultaneous embedding of pairs of planar graphs on the O(n 2 )×O(n 2 ) grid, with at most three bends per edge, where n is the number of vertices.  ...  We also describe an O(n) time algorithm for simultaneous embedding with fixed-edges for tree-path pairs on the O(n) × O(n 2 ) grid with at most one bend per tree-edge and no bends along path edges.  ...  Acknowledgments We would like to thank Michael Kaufmann and David Eppstein for the interesting discussions about the problems discussed in the paper, and Petr Moravsky for helping with the implementation  ... 
doi:10.1007/978-3-540-31843-9_21 fatcat:335gpmzsijalddsrbxfbfw426u

Simultaneous Embedding of Planar Graphs with Few Bends

Cesim Erten, Stephen G. Kobourov
2005 Journal of Graph Algorithms and Applications  
We present an O(n) time algorithm for simultaneous embedding of pairs of planar graphs on the O(n 2 )×O(n 2 ) grid, with at most three bends per edge, where n is the number of vertices.  ...  We also describe an O(n) time algorithm for simultaneous embedding with fixed-edges for tree-path pairs on the O(n) × O(n 2 ) grid with at most one bend per tree-edge and no bends along path edges.  ...  Acknowledgments We would like to thank Michael Kaufmann and David Eppstein for the interesting discussions about the problems discussed in the paper, and Petr Moravsky for helping with the implementation  ... 
doi:10.7155/jgaa.00113 fatcat:jnloyknqozbxzkf4a6wlmxeadq

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
We study the relations of these beyond-planar graph classes to right-angle crossing (RAC) graphs that admit compact drawings on the grid with few bends.  ...  Finally, we make two known algorithms embedding-preserving; for drawing 1-planar RAC graphs with at most one bend per edge and for drawing IC-planar RAC graphs straight-line.  ...  We start with the drawing of the top-level graph.  ... 
arXiv:1806.10044v4 fatcat:k2ysur7abfevdnkplhpxziguga

Lattices contained in planar orders are planar

Richard Nowakowski, Ivan Rival, Jorge Urrutia
1992 Algebra Universalis  
The covering graph of a lattice which is contained in an ordered set with a planar covering graph is itself planar.  ...  Let P(m,n) be the ordered set with m levels and with n elements at each level such that each consecutive pair of levels consists of a 2n-cycle (cf. Figure 10 ).  ...  Figure 7 We may furthermore draw this same diagram on a vertical cylinder such that the two vertical chains form parallel stripes on either side of the cylinder, that the edges with positive slope are  ... 
doi:10.1007/bf01190783 fatcat:gcirgbz6lve47psnz4oumndx64
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