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Planar Octilinear Drawings with One Bend Per Edge

Michael A. Bekos, Martin Gronemann, Michael Kaufmann, Robert Krug
2015 Journal of Graph Algorithms and Applications  
In particular, we prove that every 4-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  ... 
doi:10.7155/jgaa.00369 fatcat:hq6vjy3p3bagvdjsq3wfc7vn4y

Planar Octilinear Drawings with One Bend Per Edge [chapter]

Michael A. Bekos, Martin Gronemann, Michael Kaufmann, Robert Krug
2014 Lecture Notes in Computer Science  
In particular, we prove that every 4-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.  ... 
doi:10.1007/978-3-662-45803-7_28 fatcat:fqvopha3yjbndksu6mrommtnla

Planar Octilinear Drawings with One Bend Per Edge [article]

Michael A. Bekos, Martin Gronemann, Michael Kaufmann, Robert Krug
2014 arXiv   pre-print
In particular, we prove that every 4-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?  ... 
arXiv:1408.5920v1 fatcat:7xvbs4djxrghtev4z65igwrboq

On the Total Number of Bends for Planar Octilinear Drawings [article]

Michael A. Bekos, Michael Kaufmann, Robert Krug
2015 arXiv   pre-print
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 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.  ... 
arXiv:1512.04866v1 fatcat:qimlxodgbrh43owm473uo4tjfa

On the Total Number of Bends for Planar Octilinear Drawings [chapter]

Michael A. Bekos, Michael Kaufmann, Robert Krug
2016 Lecture Notes in Computer Science  
Since the problem of finding 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] .  ... 
doi:10.1007/978-3-662-49529-2_12 fatcat:tcndafsibjgqpl6j6slnxno3vy

On Smooth Orthogonal and Octilinear Drawings: Relations, Complexity and Kandinsky Drawings [article]

Michael A. Bekos, Henry Förster, Michael Kaufmann
2017 arXiv   pre-print
For 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.  ... 
arXiv:1708.09197v1 fatcat:ws6cfn2q4bhmne66dbcyxiisgu

On Smooth Orthogonal and Octilinear Drawings: Relations, Complexity and Kandinsky Drawings [chapter]

Michael A. Bekos, Henry Förster, Michael Kaufmann
2018 Lecture Notes in Computer Science  
For 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.  ... 
doi:10.1007/978-3-319-73915-1_15 fatcat:kklzjrle5bcltipsyeokch23ru

Universal Slope Sets for 1-Bend Planar Drawings [article]

Patrizio Angelini, Michael A. Bekos, Giuseppe Liotta, Fabrizio Montecchiani
2017 arXiv   pre-print
By universal we mean that every 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.  ... 
arXiv:1703.04283v1 fatcat:zpq54axrjvgvll2fb4v2ulin4y

Slanted Orthogonal Drawings [chapter]

Michael A. Bekos, Michael Kaufmann, Robert Krug, Stefan Näher, Vincenzo Roselli
2013 Lecture Notes in Computer Science  
While in traditional orthogonal 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  ... 
doi:10.1007/978-3-319-03841-4_37 fatcat:5stn4riwybebxo2zyzojznwgxe

Universal Slope Sets for Upward Planar Drawings [article]

Michael A. Bekos, Emilio Di Giacomo, Walter Didimo, Giuseppe Liotta, Fabrizio Montecchiani
2018 arXiv   pre-print
1-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.  ... 
arXiv:1803.09949v2 fatcat:odrwbrgyp5bnxixqafded53dcy

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.  ...  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.  ... 
doi:10.4230/dagrep.6.11.35 dblp:journals/dagstuhl-reports/Hong0KP16 fatcat:gmft5jywxbapza52lhj4o23ngq

Slanted Orthogonal Drawings: Model, Algorithms and Evaluations

Michael A. Bekos, Michael Kaufmann, Robert Krug, Thorsten Ludwig, Stefan Näher, Vincenzo Roselli
2014 Journal of Graph Algorithms and Applications  
While in traditional orthogonal 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.  ... 
doi:10.7155/jgaa.00332 fatcat:exwopfk62bhe3ar35455wbonya

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.  ...  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.  ... 
arXiv:2106.02839v2 fatcat:t62riog4n5hlbknlbe3v3h46uy

Towards Data-Driven Multilinear Metro Maps [article]

Soeren Nickel, Martin Nöllenburg
2020 arXiv   pre-print
Traditionally, most schematic metro maps in practice as well as metro map layout algorithms adhere to an octolinear layout style 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.  ... 
arXiv:1904.03039v2 fatcat:pbvk7uwjvjggbeozj35qqmztj4

Circular-Arc Cartograms [article]

Jan-Hinrich Kämper, Stephen G. Kobourov, Martin Nöllenburg
2012 arXiv   pre-print
The countries in circular-arc cartograms have the aesthetically pleasing appearance of clouds or snowflakes, depending 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.  ... 
arXiv:1112.4626v2 fatcat:4a4dvl466jfebosezcsphwxjwa
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