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Planar Octilinear Drawings with One Bend Per Edge

2015
*
Journal of Graph Algorithms and Applications
*

In particular, we prove that every 4-

doi:10.7155/jgaa.00369
fatcat:hq6vjy3p3bagvdjsq3wfc7vn4y
*planar*graph admits a*planar**octilinear**drawing**with*at most*one**bend**per**edge**on*an integer grid of size O(n 2 ) × O(n). ... For 5-*planar*graphs, we prove that*one**bend**per**edge*still suffices in order to construct*planar**octilinear**drawings*, but in super-polynomial area. ... Acknowledgement: The authors would like to thank Findan Eisenhut who implemented the algorithms for the triconnected 4and 5-*planar*graphs and reduced the running time of the 5-*planar*triconnected algorithm ...##
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Planar Octilinear Drawings with One Bend Per Edge
[chapter]

2014
*
Lecture Notes in Computer Science
*

In particular, we prove that every 4-

doi:10.1007/978-3-662-45803-7_28
fatcat:fqvopha3yjbndksu6mrommtnla
*planar*graph admits a*planar**octilinear**drawing**with*at most*one**bend**per**edge**on*an integer grid of size O(n 2 )×O(n). ... For 5-*planar*graphs, we prove that*one**bend**per**edge*still suffices in order to construct*planar**octilinear**drawings*, but in super-polynomial area. ... We prove that every 4-*planar*graph admits a*planar**octilinear**drawing**with*at most*one**bend**per**edge*in cubic area. ...##
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Planar Octilinear Drawings with One Bend Per Edge
[article]

2014
*
arXiv
*
pre-print

In particular, we prove that every 4-

arXiv:1408.5920v1
fatcat:7xvbs4djxrghtev4z65igwrboq
*planar*graph admits a*planar**octilinear**drawing**with*at most*one**bend**per**edge**on*an integer grid of size O(n^2) × O(n). ... For 5-*planar*graphs, we prove that*one**bend**per**edge*still suffices in order to construct*planar**octilinear**drawings*, but in super-polynomial area. ... • Does any triangle-free 6-*planar*graph admit a*planar**octilinear**drawing**with*at most*one**bend**per**edge*? ...##
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On the Total Number of Bends for Planar Octilinear Drawings
[article]

2015
*
arXiv
*
pre-print

An

arXiv:1512.04866v1
fatcat:qimlxodgbrh43owm473uo4tjfa
*octilinear**drawing*of a*planar*graph is*one*in which each*edge*is drawn as a sequence of horizontal, vertical and diagonal at 45 degrees line-segments. ... As the problem of finding*planar**octilinear**drawings*of minimum number of*bends*is NP-hard, in this paper we focus*on*upper and lower bounds. ... Note that there exist 6-*planar*graphs that do not admit*planar**octilinear**drawings**with*at most*one**bend**per**edge*[2] . Theorem 3 implies that*on*average*one**bend**per**edge*suffices. ...##
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On the Total Number of Bends for Planar Octilinear Drawings
[chapter]

2016
*
Lecture Notes in Computer Science
*

Since the problem of finding

doi:10.1007/978-3-662-49529-2_12
fatcat:tcndafsibjgqpl6j6slnxno3vy
*planar**octilinear**drawings**with*minimum number of*bends*is NP-hard, in this paper we focus*on*upper and lower bounds. ... An*octilinear**drawing*of a*planar*graph is*one*in which each*edge*is drawn as a sequence of horizontal, vertical and diagonal at 45 • and −45 • line-segments. ... Recently, it was proved that all*planar*graphs of maximum degree 4 and 5 admit*planar**octilinear**drawings**with*at most*one**bend**per**edge*[2] . ...##
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On Smooth Orthogonal and Octilinear Drawings: Relations, Complexity and Kandinsky Drawings
[article]

2017
*
arXiv
*
pre-print

For

arXiv:1708.09197v1
fatcat:ws6cfn2q4bhmne66dbcyxiisgu
*planar*graphs of higher degree, we present an algorithm that produces bi-monotone smooth orthogonal*drawings**with*at most two segments*per**edge*, which also guarantees a linear number of*edges**with*... We study two variants of the well-known orthogonal*drawing*model: (i) the smooth orthogonal, and (ii) the*octilinear*. ... We refer to*drawings**with**edge*complexity k as k-*drawings*; thus, by definition, orthogonal k-*drawings*have at most k − 1*bends**per**edge*. Known results. ...##
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On Smooth Orthogonal and Octilinear Drawings: Relations, Complexity and Kandinsky Drawings
[chapter]

2018
*
Lecture Notes in Computer Science
*

For

doi:10.1007/978-3-319-73915-1_15
fatcat:kklzjrle5bcltipsyeokch23ru
*planar*graphs of higher degree, we present an algorithm that produces bi-monotone smooth orthogonal*drawings**with*at most two segments*per**edge*, which also guarantees a linear number of*edges**with*... We study two variants of the well-known orthogonal*drawing*model: (i) the smooth orthogonal, and (ii) the*octilinear*. ... We refer to*drawings**with**edge*complexity k as k-*drawings*; thus, by definition, orthogonal k-*drawings*have at most k − 1*bends**per**edge*. Known results. ...##
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Universal Slope Sets for 1-Bend Planar Drawings
[article]

2017
*
arXiv
*
pre-print

By universal we mean that every

arXiv:1703.04283v1
fatcat:zpq54axrjvgvll2fb4v2ulin4y
*planar*graph of degree Δ has a*planar**drawing**with*at most*one**bend**per**edge*and such that the slopes of the segments forming the*edges*belong to the given set of slopes ... We describe a set of Δ -1 slopes that are universal for 1-*bend**planar**drawings*of*planar*graphs of maximum degree Δ≥ 4; this establishes a new upper bound of Δ-1*on*the 1-*bend**planar*slope number. ... This work started at the 19 th Korean Workshop*on*Computational Geometry. ...##
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Slanted Orthogonal Drawings
[chapter]

2013
*
Lecture Notes in Computer Science
*

While in traditional orthogonal

doi:10.1007/978-3-319-03841-4_37
fatcat:5stn4riwybebxo2zyzojznwgxe
*drawings*each*edge*is made of axis-aligned line-segments, in slanted orthogonal*drawings*intermediate diagonal segments*on*the*edges*are also permitted, which allows for ...*On*the negative side, we show that*bend*-optimal slanted orthogonal*drawings*may require exponential area. ... Given a slog representation of a*planarized*graph G*with*maxdegree 4, we can efficiently compute a slog*drawing*requiring O(n 2 ) area*with*(i) optimal number of half-*bends**on*rr*edges*and rc*edges*without ...##
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Universal Slope Sets for Upward Planar Drawings
[article]

2018
*
arXiv
*
pre-print

1-

arXiv:1803.09949v2
fatcat:odrwbrgyp5bnxixqafded53dcy
*bend*upward*planar**drawing*whose*edge*segments use only slopes in S. ... In addition, we prove that every such set S can be used to construct 2-*bend*upward*planar**drawings*of n-vertex*planar*st-graphs*with*at most 4n-9*bends*in total. ... Upward*drawings**with**one**bend**per**edge*and few slopes have also been studied for posets by Czyzowicz et al. [10] . Contribution. ...##
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Beyond-Planar Graphs: Algorithmics and Combinatorics (Dagstuhl Seminar 16452)

2017
*
Dagstuhl Reports
*

The common interest was in the exploration of structural properties and the development of algorithms for so-called beyond-

doi:10.4230/dagrep.6.11.35
dblp:journals/dagstuhl-reports/Hong0KP16
fatcat:gmft5jywxbapza52lhj4o23ngq
*planar*graphs, i.e., non-*planar*graphs*with*topological constraints such as specific ... types of crossings, or*with*some forbidden crossing patterns. ... For the case that the given graph is*planar*, they describe how to obtain a*planar*orthogonal*drawing**with*at most two*bends**per**edge*, except possibly for*one**edge**on*the outer face. ...##
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Slanted Orthogonal Drawings: Model, Algorithms and Evaluations

2014
*
Journal of Graph Algorithms and Applications
*

While in traditional orthogonal

doi:10.7155/jgaa.00332
fatcat:exwopfk62bhe3ar35455wbonya
*drawings*each*edge*is made of alternating axis-aligned line segments, in slanted orthogonal*drawings*intermediate diagonal segments*on*the*edges*are permitted, which allows ...*On*the negative side, we show that*bend*-optimal slanted orthogonal*drawings*may require exponential area. ... Part of the research was conducted in the framework of ESF project 10-EuroGIGA-OP-003 GraDR "Graph*Drawings*and Representations". The work of M.A. ...##
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Upward planar drawings with two slopes
[article]

2021
*
arXiv
*
pre-print

As an application of this

arXiv:2106.02839v2
fatcat:t62riog4n5hlbknlbe3v3h46uy
*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. ... An -*bend*k-slope*drawing*of G is an -*bend**drawing*of G where every*edge*segment has*one*of at most k distinct slopes. ...##
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Towards Data-Driven Multilinear Metro Maps
[article]

2020
*
arXiv
*
pre-print

Traditionally, most schematic metro maps in practice as well as metro map layout algorithms adhere to an octolinear layout style

arXiv:1904.03039v2
fatcat:pbvk7uwjvjggbeozj35qqmztj4
*with*all paths composed of horizontal, vertical, and 45-diagonal*edges*. ... Despite growing interest in more general multilinear metro maps, generic algorithms to*draw*metro maps based*on*a system of k > 2 not necessarily equidistant slopes have not been investigated thoroughly ... The measures are the number of*bends*, sector deviation (total and*per**edge*), distortion*per**edge*and the runtime in seconds. ...##
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Circular-Arc Cartograms
[article]

2012
*
arXiv
*
pre-print

The countries in circular-arc cartograms have the aesthetically pleasing appearance of clouds or snowflakes, depending

arXiv:1112.4626v2
fatcat:4a4dvl466jfebosezcsphwxjwa
*on*whether their*edges*are bent outwards or inwards. ... Next we describe a heuristic method for constructing circular-arc cartograms, which uses a max-flow computation*on*the dual graph of the map, along*with*a computation of the straight skeleton of the underlying ... Fig. 2 : 2 A*planar*rectilinear*drawing*of a*PLANAR*MONOTONE 3-SAT instance. Fig. 3 : 3 The three possibilities to realize a circular-arc triangle*with*area 0. Fig. 4 : 4 Variable gadget. ...
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