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Simultaneous Embedding of a Planar Graph and Its Dual on the Grid
[article]

2002
*
arXiv
*
pre-print

We provide

arXiv:cs/0206019v1
fatcat:aql4xmkubnhvfjxygzyurt5mfq
*a*linear-time algorithm that*simultaneously*embeds*a*3-connected*planar**graph**and**its**dual**on**a*(2n-2) by (2n-2) integer*grid*, where n is*the*total number*of*vertices in*the**graph**and**its**dual*... We consider*the*problem*of**simultaneously**embedding**a**planar**graph**and**its**dual*into*a*small integer*grid*such that*the*edges are drawn as straight-line segments*and**the*only crossings are between primal-*dual*... Introduction In this paper we address*the*problem*of**simultaneously*drawing*a**planar**graph**and**its**dual**on**a*small integer*grid*. ...##
###
Simultaneous Embedding of a Planar Graph and Its Dual on the Grid
[chapter]

2002
*
Lecture Notes in Computer Science
*

We consider

doi:10.1007/3-540-36136-7_50
fatcat:fgtexougqncfbiklwrh5jyssse
*the*problem*of**simultaneously**embedding**a**planar**graph**and**its**dual**on**a*small integer*grid*such that*the*edges are drawn as straight-line segments*and**the*only crossings are between primal-*dual*... We provide an O(n) time algorithm that*simultaneously*embeds*a*3-connected*planar**graph**and**its**dual**on**a*(2n − 2) × (2n − 2) integer*grid*, where n is*the*total number*of*vertices in*the**graph**and**its*... Introduction In this paper we address*the*problem*of**simultaneously*drawing*a**planar**graph**and**its**dual**on**a*small integer*grid*. ...##
###
Simultaneous Embedding of a Planar Graph and Its Dual on the Grid

2005
*
Theory of Computing Systems
*

We consider

doi:10.1007/s00224-005-1143-4
fatcat:wgvjotnk2rarnpjrjimkrl5ixq
*the*problem*of**simultaneously**embedding**a**planar**graph**and**its**dual**on**a*small integer*grid*such that*the*edges are drawn as straight-line segments*and**the*only crossings are between primal-*dual*... We provide an O(n) time algorithm that*simultaneously*embeds*a*3-connected*planar**graph**and**its**dual**on**a*(2n − 2) × (2n − 2) integer*grid*, where n is*the*total number*of*vertices in*the**graph**and**its*... Introduction In this paper we address*the*problem*of**simultaneously*drawing*a**planar**graph**and**its**dual**on**a*small integer*grid*. ...##
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On Simultaneous Graph Embedding
[article]

2002
*
arXiv
*
pre-print

We consider

arXiv:cs/0206018v3
fatcat:xrrjocicsjekhdfddudvpok5sq
*the*problem*of**simultaneous**embedding**of**planar**graphs*. ... In particular, we show that without mapping, any number*of*outerplanar*graphs*can be*embedded**simultaneously**on*an O(n)× O(n)*grid*,*and*an outerplanar*and*general*planar**graph*can be*embedded**simultaneously*... Acknowledgments We would like to thank Anna Lubiw for introducing us to*the*problem*of**simultaneous**graph**embedding*. ...##
###
On simultaneous planar graph embeddings

2007
*
Computational geometry
*

We consider

doi:10.1016/j.comgeo.2006.05.006
fatcat:67uq7tu57nhafasdmca74biuam
*the*problem*of**simultaneous**embedding**of**planar**graphs*. ... If*the*mapping is not given, we show that any number*of*outerplanar*graphs*can be*embedded**simultaneously**on*an O(n) × O(n)*grid*,*and*an outerplanar*and*general*planar**graph*can be*embedded**simultaneously*... Acknowledgments We would like to thank Ed Scheinerman for stimulating discussions about different variations*of**the*problem*and*Esther M. Arkin for her proof*of*Theorem 3 (independent*of*our work). ...##
###
On Simultaneous Planar Graph Embeddings
[chapter]

2003
*
Lecture Notes in Computer Science
*

We consider

doi:10.1007/978-3-540-45078-8_22
fatcat:6sbtuz47offrtda4venwyieb3a
*the*problem*of**simultaneous**embedding**of**planar**graphs*. ... If*the*mapping is not given, we show that any number*of*outerplanar*graphs*can be*embedded**simultaneously**on*an O(n) × O(n)*grid*,*and*an outerplanar*and*general*planar**graph*can be*embedded**simultaneously*... Erten*and*Kobourov [13] present an algorithm for*simultaneously**embedding**a*3-connected*planar**graph**and**its**dual**on*an O(n) × O(n)*grid*. ...##
###
Simultaneous Embedding of Planar Graphs
[article]

2015
*
arXiv
*
pre-print

Recently there has been

arXiv:1204.5853v3
fatcat:op5qpoub7bfhlgljxqygzx6jha
*a*great deal*of*work investigating*simultaneous**embedding*problems both from*a*theoretical*and**a*practical point*of*view. We survey recent work*on*this topic. ...*Simultaneous**embedding*is concerned with*simultaneously*representing*a*series*of**graphs*sharing some or all vertices. ...*A*Plane*Graph**and**its**Dual*In*a**simultaneous*drawing*of**a**planar**graph**and**its**dual*each vertex in*the**dual**graph*is required to be placed inside*the*corresponding face*of**the*primal*graph*. ...##
###
Combining Problems on RAC Drawings and Simultaneous Graph Drawings
[chapter]

2012
*
Lecture Notes in Computer Science
*

This is summarized in

doi:10.1007/978-3-642-25878-7_41
fatcat:62icfokbwnfgded6ktmeipfb6a
*the*following theorem. Theorem 4. Given*a**planar**embedded**graph*G,*a*GDual-GRacSim drawing*of*G*and**its**dual*G * does not always exist. ...*A*closely related problem is*the*following: Given*a**planar**embedded**graph*G, determine*a*geometric drawing*of*G*and**its**dual*G * (without*the*face-vertex corresponding to*the*external face) such that: ...##
###
Simultaneous Embeddability of Two Partitions
[article]

2014
*
arXiv
*
pre-print

We show that (i) every pair

arXiv:1408.6019v1
fatcat:ffxzjrahcbdndalpqko45c3yzi
*of*partitions has*a*weak*simultaneous**embedding*, (ii)*it*is NP-complete to decide*the*existence*of**a*strong*simultaneous**embedding*,*and*(iii)*the*existence*of**a*full*simultaneous*... We study*the**simultaneous*embeddability*of**a*pair*of*partitions*of**the*same underlying set into disjoint blocks. ... We call G Γ*the*contour*graph**of*Γ*and**its**dual**graph*G * Γ*the**dual**graph**of*Γ .*The*faces*of*G Γ belong to zero,*one*, or two block regions. ...##
###
Simultaneous Embeddability of Two Partitions
[chapter]

2014
*
Lecture Notes in Computer Science
*

We show that (i) every pair

doi:10.1007/978-3-662-45803-7_6
fatcat:4qflfetdxre7tpsafwf3lmi2qm
*of*partitions has*a*weak*simultaneous**embedding*, (ii)*it*is NP-complete to decide*the*existence*of**a*strong*simultaneous**embedding*,*and*(iii)*the*existence*of**a*full*simultaneous*... We study*the**simultaneous*embeddability*of**a*pair*of*partitions*of**the*same underlying set into disjoint blocks. ... We thank*the*anonymous reviewers for helpful comments. References ...##
###
Geometric RAC Simultaneous Drawings of Graphs

2013
*
Journal of Graph Algorithms and Applications
*

goal is to determine

doi:10.7155/jgaa.00282
fatcat:e4sm3vhimzdbzf53lkttygt2hm
*a*geometric drawing*of*G*and**its*weak*dual*G * such that: (i) G*and*G * are drawn*planar*, (ii) each vertex*of**the**dual*is drawn inside*its*corresponding face*of*G*and*, (iii)*the*primal-*dual*... We further show that*a*cycle*and**a*matching always admit*a*geometric RAC*simultaneous*drawing.We also study*a*closely related problem according to which we are given*a**planar**embedded**graph*G*and**the*main ...*The*work*of*M.A. ...##
###
Geometric RAC Simultaneous Drawings of Graphs
[chapter]

2012
*
Lecture Notes in Computer Science
*

goal is to determine

doi:10.1007/978-3-642-32241-9_25
fatcat:uor4ui6q3jhhpcdyaylwwezoqi
*a*geometric drawing*of*G*and**its*weak*dual*G * such that: (i) G*and*G * are drawn*planar*, (ii) each vertex*of**the**dual*is drawn inside*its*corresponding face*of*G*and*, (iii)*the*primal-*dual*... We further show that*a*cycle*and**a*matching always admit*a*geometric RAC*simultaneous*drawing.We also study*a*closely related problem according to which we are given*a**planar**embedded**graph*G*and**the*main ...*The*work*of*M.A. ...##
###
Monotone Simultaneous Embeddings of Upward Planar Digraphs

2015
*
Journal of Graph Algorithms and Applications
*

We study monotone

doi:10.7155/jgaa.00350
fatcat:46tdmqjuovdj7i2gzg3qucpdce
*simultaneous**embeddings**of*upward*planar*digraphs, which are*simultaneous**embeddings*where*the*drawing*of*each digraph is upward*planar*,*and**the*directions*of**the*upwardness*of*different ...*On**the*other hand, we show that already for three paths, any monotone*simultaneous**embedding*might need*a**grid*whose size is exponential in*the*number*of*vertices. ... Acknowledgements We thank anonymous referees for helpful comments*and*for making us aware*of**the*related work [3] . ...##
###
Monotone Simultaneous Embedding of Directed Paths
[article]

2014
*
arXiv
*
pre-print

We study monotone

arXiv:1310.6955v2
fatcat:dpvxxprvpfhnrlepbnpzczmur4
*simultaneous**embeddings**of*upward*planar*digraphs, which are*simultaneous**embeddings*where*the*drawing*of*each digraph is upward*planar*,*and**the*directions*of**the*upwardness*of*different ...*On**the*other hand, we show that already for three paths, any monotone*simultaneous**embedding*might need*a**grid**of*exponential (w.r.t.*the*number*of*vertices) size. ... Research*on*this topic was initiated during*a*research visit*of*Sarah Lutteropp*and*Tamara Mchedlidze in February 2013 in Graz, Austria. ...##
###
Level-Planar Drawings with Few Slopes
[article]

2019
*
arXiv
*
pre-print

Moreover, we consider

arXiv:1907.13558v1
fatcat:a6mtdmeqgfdbvlmvgjn4nm323y
*the*partial drawing extension problem, where we seek to extend an immutable drawing*of**a*subgraph to*a*drawing*of**the*whole*graph*,*and**the**simultaneous*drawing problem, which asks ... We introduce*and*study level-*planar*straight-line drawings with*a*fixed number λ*of*slopes. ...*It*is evident that any level-*planar*two-slope*grid*drawing*of**the*resulting*graph*induces*a*solution*of**the*underlying*planar*monotone 3-Sat problem*and*vice versa. ...
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