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A Note on Minimum-Segment Drawings of Planar Graphs

2013
*
Journal of Graph Algorithms and Applications
*

*A*straight-line

*drawing*

*of*

*a*

*planar*

*graph*G is

*a*

*planar*

*drawing*

*of*G, where each vertex is mapped to

*a*point

*on*the Euclidean plane and each edge is drawn as

*a*straight line

*segment*. ... Finally, we investigate

*a*worst-case lower bound

*on*the number

*of*

*segments*required by straight-line

*drawings*

*of*arbitrary spanning trees

*of*

*a*given

*planar*

*graph*. ...

*A*natural question is: what is the time complexity

*of*computing

*a*

*minimum*-

*segment*

*drawing*

*of*

*a*

*planar*

*graph*[2] ? Dujmović et al. ...

##
###
Area Minimization for Grid Visibility Representation of Hierarchically Planar Graphs
[chapter]

1999
*
Lecture Notes in Computer Science
*

Firstly, we provide

doi:10.1007/3-540-48686-0_9
fatcat:u3mbk5b4ejegpabyxbttnnhgmi
*a*quadratic algorithm that minimizes the*drawing*area with respect to*a*xed*planar*embedding. ... In this paper, we shall investigate the computational complexity*of*constructing*minimum*area grid visibility representations*of*hierarchically*planar**graphs*. ... Moreover, an e cient grid visibility representation algorithm was presented that can achieve the*minimum**drawing*area for*a*hierarchically*planar**graph*with only*one*\source", only*one*\sink", and*a*xed ...##
###
The Complexity of Drawing a Graph in a Polygonal Region
[article]

2018
*
arXiv
*
pre-print

Our result is

arXiv:1802.06699v3
fatcat:gkzwt5a2jrffpg5dzroymbacv4
*one**of*the first showing that*a*problem*of**drawing**planar**graphs*with straight-line edges is hard for the existential theory*of*the reals. ...*of*the region, place the remaining vertices to create*a**planar*straight-line*drawing**of*the*graph*inside the region. ... We would like to thank Günter Rote, who discussed with the second author the turn gadget in the context*of*the Art Gallery Problem. The Complexity*of**Drawing**a**Graph*in*a*Polygonal Region ...##
###
Capturing Lombardi Flow in Orthogonal Drawings by Minimizing the Number of Segments
[article]

2016
*
arXiv
*
pre-print

We study two problems

arXiv:1608.03943v1
fatcat:veivfxkabremnkur4ii556cewm
*on*orthogonal*drawings**of**planar**graphs*,*one*that minimizes the total number*of*line*segments*and another that minimizes the number*of*line*segments*that cover all the vertices. ... For both*graph*classes, we give polynomial-time algorithms for upward orthogonal*drawings*with the*minimum*number*of**segments*covering the vertices. ... This article reports*on*work supported by the Defense Advanced Research Projects Agency under agreement no. AFRL FA8750-15-2-0092. ...##
###
Towards area requirements for drawing hierarchically planar graphs

2003
*
Theoretical Computer Science
*

An application

doi:10.1016/s0304-3975(02)00031-2
fatcat:llocylvu55hl5amntzu35n7lrq
*of*the existing results from upward*drawing*can guarantee*a*quadric*drawing*area for grid visibility representation but does not necessarily guarantee the*minimum**drawing*area. ... Motivated by this, we will present*a*new grid visibility*drawing*algorithm which is e cient and guarantees the*minimum**drawing*area with respect to*a*given topological embedding. ... Acknowledgements The authors are very grateful to the anonymous referees for*a*careful review and many valuable comments for improving the presentation*of*the paper. ...##
###
Minimum Length Embedding of Planar Graphs at Fixed Vertex Locations
[chapter]

2013
*
Lecture Notes in Computer Science
*

We consider the problem

doi:10.1007/978-3-319-03841-4_33
fatcat:gj6aw4zwybf2hjgix2jcctgohu
*of*finding*a**planar*embedding*of**a**graph*at fixed vertex locations that minimizes the total edge length. The problem is known to be NP-hard. ... We give polynomial time algorithms achieving an O( √ n log n) approximation for paths and matchings, and an O(n) approximation for general*graphs*. ... We learned about Sam Loyd's disjoint paths puzzle (which is not original to him) from Marcus Schaefer who has studied the history*of*such*planarity*puzzles. ...##
###
On the Minimum Cut of Planarizations

2007
*
Electronic Notes in Discrete Mathematics
*

Every

doi:10.1016/j.endm.2007.01.036
fatcat:cxsebbtjlne2rkhnacydltmizi
*drawing**of**a*non-*planar**graph*G in the plane induces*a**planarization*, i.e.,*a**planar**graph*obtained by replacing edge crossings with dummy vertices. ...*On*the other hand, we prove that every crossing minimal*planarization*can be efficiently transformed into another crossing minimal*planarization*that preserves the capacity*of**a**minimum*st-cut in G. ... In this paper, we deal with*graph*theoretic properties*of*the*planar**graph*induced by*a**drawing**of*G. ...##
###
Area requirements for drawing hierarchically planar graphs
[chapter]

1997
*
Lecture Notes in Computer Science
*

Two

doi:10.1007/3-540-63938-1_64
fatcat:vqxhepwm2bec5p2vamstciax4u
*drawing*standards will be discussed: 1) each vertex is represented by*a*point and 2) grid visibifity representation (that is,*a*line*segment*is allowed to represent*a*vertex). ... Applications*of*some existing algorithms from upward*drawing*can guarantee the quadratic*drawing*area for grid visibility representation but do not necessarily guarantee the*minimum**drawing*area. ... Respecting*a**planar*embedding EH*of**a*hierarchically*planar**graph*H, the grid visibility representation*of*H produced by the algorithm GVP has the*minimum**drawing*area. ...##
###
Obstacle Numbers of Planar Graphs
[article]

2017
*
arXiv
*
pre-print

If G is

arXiv:1706.06992v3
fatcat:ucgmzezbazd6rhqwfborctcjtq
*planar*, we define the*planar*obstacle number*of*G by further requiring that the visibility*graph*has no crossing edges (hence that it is*a**planar*geometric*drawing**of*G). ...*graphs**of*straight line*segments*in two directions)*of*order at least 3 is 1. ... Acknowledgments We thank Vít Jelínek for ideas leading to*a*simplification*of*our proof. We also thank Daniel Král ' and Roman Nedela for inspiring comments*on*our research. ...##
###
Drawing Outer-Planar Graphs in O(n log n )Area
[chapter]

2002
*
Lecture Notes in Computer Science
*

In this paper, we study

doi:10.1007/3-540-36151-0_6
fatcat:n67ueprnbvbn3ef4h3btbir5va
*drawings**of*outer-*planar**graphs*in various models. ... The question*of*straight-line grid-*drawings**of*outerplanar*graphs*in o(n 2 ) area remains open. ... There exists an outer-*planar**graph*G such that any poly-line*drawing*Γ*of*G with all vertices*on*the boundary*of*the*minimum*enclosing rectangle*of*the*graph*has area Ω(n 2 ). Proof. ...##
###
Orthogonal drawings of graphs with vertex and edge labels

2005
*
Computational geometry
*

This paper studies the problem

doi:10.1016/j.comgeo.2005.02.001
fatcat:whimtnqjq5f4pcp6fx6l7f2o4q
*of*computing orthogonal*drawings**of**graphs*with labels*on*vertices and edges. ... We present MILP models to compute optimal*drawings*with respect to the first three goals and an exact algorithm that is based*on*these models to compute*a*labeled*drawing**of**minimum*area. ... Orthogonal*drawings**A**planar*orthogonal*drawing*Γ*of**a**planar**graph*G is*a**planar**drawing**of*G such that each edge e is mapped to*a*sequence*of*horizontal and vertical*segments*:*a*left or*a*right turn ...##
###
Slanted Orthogonal Drawings
[chapter]

2013
*
Lecture Notes in Computer Science
*

We introduce

doi:10.1007/978-3-319-03841-4_37
fatcat:5stn4riwybebxo2zyzojznwgxe
*a*new model that we call slanted orthogonal*graph**drawing*. ... 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 ...*Note*that the general problem*of*determining*a**planar*embedding with the*minimum*number*of*bends is NP-hard [5] , which is also the case for slog*drawings*. ...##
###
Really Straight Graph Drawings
[chapter]

2005
*
Lecture Notes in Computer Science
*

We prove that every 3-connected plane

doi:10.1007/978-3-540-31843-9_14
fatcat:fwradttc6ba7jip6ftrdahsdwq
*graph**on*n vertices has*a*plane*drawing*with at most 5n/2*segments*and at most 2n slopes, and that every cubic 3-connected plane*graph*has*a*plane*drawing*with three ... 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. ... Acknowledgements Thanks to all*of*the participants*of*the Bellairs workshop for creating*a*stimulating working environment. Special thanks to Mike Fellows for suggesting the problem. ...##
###
Smooth Orthogonal Layouts
[chapter]

2013
*
Lecture Notes in Computer Science
*

We study the problem

doi:10.1007/978-3-642-36763-2_14
fatcat:5zg4c4omgrekrmuocaklaaqxha
*of*creating smooth orthogonal layouts for*planar**graphs*. ... Using the Kandinsky model for removing the degree restriction, we show that any*planar**graph*has*a*smooth complexity-2 layout. The work*of*M.A. ... Let G be*a*4-*planar**graph*that admits an orthogonal*drawing*H with*minimum*number*of*bends and edge complexity k. ...##
###
Beyond-Planar Graphs: Algorithmics and Combinatorics (Dagstuhl Seminar 16452)

2017
*
Dagstuhl Reports
*

The seminar brought together 29 researchers in the areas

doi:10.4230/dagrep.6.11.35
dblp:journals/dagstuhl-reports/Hong0KP16
fatcat:gmft5jywxbapza52lhj4o23ngq
*of**graph*theory, combinatorics, computational geometry, and*graph**drawing*. ... 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 ...*A**note**on**minimum*-*segment**drawings**of**planar**graphs*. J.*Graph*Algorithms Appl., 17:301-328, 2013. 14 David Eppstein, Marc van Kreveld, Elena Mumford, and Bettina Speckmann. ...
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