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Planarizing Graphs - A Survey and Annotated Bibliography

2001
*
Journal of Graph Algorithms and Applications
*

While there are many

doi:10.7155/jgaa.00032
fatcat:zftlx3a5jnesxff5hjymjewp5u
*algorithmic*results about*planarization*through*edge*deletion, the results about vertex*splitting*, thickness, and crossing number are mostly of a structural nature. ... Given a finite, undirected, simple*graph*G, we are concerned with operations on G that transform it into a*planar**graph*. ... H, S, and K define*an*instance of Eligible Set*Split**Planar**Graph*. G has a*planar*subgraph with K or more*edges*if and only if H can be*planarized*by K vertex*splitting*operations on S. ...##
###
A Fixed-Parameter Algorithm for the Max-Cut Problem on Embedded 1-Planar Graphs
[chapter]

2018
*
Lecture Notes in Computer Science
*

Our

doi:10.1007/978-3-319-94667-2_12
fatcat:4nkl37qqmrbulljobuvvuiiiy4
*algorithm*computes a maximum cut*in**an*embedded 1-*planar**graph*with n nodes and k*edge*crossings*in*time O(3^k · n^3/2 n). ... Our*algorithm*recursively reduces a 1-*planar**graph*to at most 3^k*planar**graphs*, using*edge*removal and node contraction. ...*Split*(S2, vyz) 14: else 15: S ← S3 16: end if 17: end if 18: return S*Algorithm*1.1: Max-Cut*algorithm*for embedded 1-*planar**graphs**an*arbitrary crossing is selected and removed*in*three different ways ...##
###
Inserting an Edge into a Planar Graph

2005
*
Algorithmica
*

Computing a crossing minimum drawing of a given

doi:10.1007/s00453-004-1128-8
fatcat:pteako5g6zhe3o7k3cfqopvw2a
*planar**graph*G augmented by*an*additional*edge*e*in*which all crossings involve e, has been a long standing open problem*in**graph*drawing. ... Alternatively, the problem can be stated as finding a*planar*combinatorial embedding of a*planar**graph*G*in*which the given*edge*e can be inserted with the minimum number of crossings. ...*Algorithm*1 computes*an*optimal*edge*insertion path for a biconnected*planar**graph*and two nonadjacent vertices*in*this*graph*. ...##
###
Bitonic st-orderings for Upward Planar Graphs
[article]

2016
*
arXiv
*
pre-print

We fully characterize

arXiv:1608.08578v1
fatcat:lwghubpmhnfzhipvkqebheokpq
*planar*st-*graphs*that admit such*an*ordering and provide a linear-time*algorithm*for recognition and ordering. ... If for a*graph*no bitonic st-ordering exists, we show how to find*in*linear time a minimum set of*edges*to*split*such that the resulting*graph*admits one. ... a path*Algorithm*2:*Algorithm*for computing the minimum set of*edges*to*split*. input : Embedded*planar*st-*graph*G = (V, E) with S(u) for every u ∈ V . output: Minimum set E*split*⊂ E to*split*for admitting ...##
###
Bitonic st-orderings for Upward Planar Graphs
[chapter]

2016
*
Lecture Notes in Computer Science
*

We fully characterize

doi:10.1007/978-3-319-50106-2_18
fatcat:cielzn3udfcpli7syegtbvgfli
*planar*st-*graphs*that admit such*an*ordering and provide a lineartime*algorithm*for recognition and ordering. ... If for a*graph*no bitonic st-ordering exists, we show how to find*in*linear time a minimum set of*edges*to*split*such that the resulting*graph*admits one. ... a path*Algorithm*2:*Algorithm*for computing the minimum set of*edges*to*split*. input : Embedded*planar*st-*graph*G = (V, E) with S(u) for every u ∈ V . output: Minimum set E*split*⊂ E to*split*for admitting ...##
###
Subgraph Induced Planar Connectivity Augmentation
[chapter]

2003
*
Lecture Notes in Computer Science
*

The problem arises

doi:10.1007/978-3-540-39890-5_23
fatcat:mrcjncqhh5fw5bztb5bm6vuriu
*in*automatic*graph*drawing*in*the context of c-*planarity*testing of clustered*graphs*. ... Given a*planar**graph*G = (V, E) and a vertex set W ⊆ V , the subgraph induced*planar*connectivity augmentation problem asks for a minimum cardinality set F of additional*edges*with end vertices*in*W such ... They represent a decomposition of a*planar*biconnected*graph*according to its*split*pairs (pairs of vertices whose removal*splits*the*graph*or vertices connected by*an**edge*). ...##
###
A Linear-Time Algorithm for Finding a Maximal Planar Subgraph

2006
*
SIAM Journal on Discrete Mathematics
*

We construct

doi:10.1137/s0895480197328771
fatcat:muvfonqy7nbgdd2hinkjpavvaq
*an*optimal linear-time*algorithm*for the maximal*planar*subgraph problem: given a*graph*G, find a*planar*subgraph G of G such that adding to G*an*extra*edge*of G results*in*a non-*planar**graph*... Our*algorithm*can be transformed into a new optimal*planarity*testing*algorithm*. ...*In*Section 3 we develop*an**algorithm*for on-line*planarity*testing*in*triconnected*graphs**in*a constant amortized time, which we use as a subroutine*in*the main*algorithm*. ...##
###
Page 7635 of Mathematical Reviews Vol. , Issue 2000k
[page]

2000
*
Mathematical Reviews
*

*In*this paper, we prove that,

*in*the case where G is

*planar*and k is

*an*even integer or k = 3, there exists a complete

*splitting*at s such that the resulting

*graph*G’ remains k-

*edge*-connected and

*planar*... It is known that all

*edges*incident to s can be

*split*without losing the

*edge*-connectivity of G

*in*V —s. This complete

*splitting*plays

*an*important role

*in*solving many

*graph*connectivity problems. ...

##
###
A new planarity test based on 3-connectivity

1970
*
IEEE Transactions on Circuit Theory
*

Initialization of

doi:10.1109/tct.1970.1083101
fatcat:jsjyquwihraihark5o25zhbbcq
*Algorithm**Planar*Let G be*an*arbitrary*graph*. ... It has been shown by Mac Lane [4] that the*graph*H is*planar*if and only if J' and K' are*planar**graphs*. This*splitting*process is a basic step*in**Algorithm**Planar*. ...##
###
On-line convex planarity testing
[chapter]

1995
*
Lecture Notes in Computer Science
*

*An*important class of

*planar*straight-line drawings of

*graphs*are the convex drawings,

*in*which all faces are drawn as convex polygons. ... We consider the problem of testing convex

*planarity*

*in*a semidynamic environment, where a

*graph*is subject to on-line insertions of vertices and

*edges*. ... Introduction Develop a fully dynamic convex

*planarity*testing

*algorithm*. The best

*algorithm*for fully dynamic

*planarity*testing performs query and update operations

*in*amortized time O( p n) 1 7 ] . ...

##
###
Graph Embedding with Minimum Depth and Maximum External Face
[chapter]

2004
*
Lecture Notes in Computer Science
*

We present new linear time

doi:10.1007/978-3-540-24595-7_24
fatcat:gjr22axtjncyjjn3phedsxnx7q
*algorithms*using the SPQR-tree data structure for computing*planar*embeddings of*planar**graphs*optimizing certain distance measures. ... Given a*planar**graph*, the*algorithms*compute the*planar*embedding with 1. the minimum depth among the set of all*planar*embeddings of G, 2. the external face of maximum size among the set of all*planar*...*Algorithms*for the second step typically deal with*planar**graphs*(i.e., the*planarized**graphs*arising*in*step 1). ...##
###
Fixed-Parameter Algorithms for the Weighted Max-Cut Problem on Embedded 1-Planar Graphs
[article]

2020
*
arXiv
*
pre-print

Our

arXiv:1812.03074v4
fatcat:zwhccl3ukncehghs6skhhhetgm
*algorithms*compute a maximum cut*in**an*embedded weighted 1-*planar**graph*with n nodes and k*edge*crossings*in*time O(3^k · n^3/2log n). ... Our*algorithms*recursively reduce a 1-*planar**graph*to at most 3^k*planar**graphs*, using*edge*removal and node contraction. ... Weighted Max-Cut*algorithm*for embedded 1-*planar**graphs*with non-negative*edge*weigths. Figure 2 : 2*An*example how the*algorithm*calculates a Max-Cut*in**an*embedded 2-almost-*planar**graph*. ...##
###
Deciding Conjugacy in Thompson's Group F in Linear Time

2013
*
2013 15th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing
*

We prove that our

doi:10.1109/synasc.2013.19
dblp:conf/synasc/HossainMBM13
fatcat:42rtnq4n7jbtplyzq4azufdyfq
*algorithm*theoretically achieves*an*O(n) bound*in*the size of the input, and we present a O(n 2 ) working solution. ... This*algorithm*checks for conjugacy by constructing and comparing directed*graphs*called strand diagrams. ... Note that their*algorithm*accepts*planar**graphs*with loops and multiple*edges*between vertices. ...##
###
Bend Minimization in Planar Orthogonal Drawings Using Integer Programming

2006
*
SIAM Journal on Optimization
*

We consider the problem of minimizing the number of bends

doi:10.1137/040614086
fatcat:q2giaw2mw5gktixad3tczybv7y
*in*a*planar*orthogonal*graph*drawing. ... Our approach for biconnected*graphs*combines*an*integer linear programming (ILP) formulation for the set of all embeddings of a*planar**graph*with the network flow formulation for fixed embeddings. ... Acknowledgment We thank Walter Didimo for providing the code of the branch & bound*algorithm*and the real world benchmark set. ...##
###
Partitioning planar graphs: a fast combinatorial approach for max-cut

2010
*
Computational optimization and applications
*

*In*this work, we present a new and simple

*algorithm*for determining maximum cuts for arbitrary weighted

*planar*

*graphs*. ... For

*planar*

*graphs*, there exist several polynomial-time methods determining maximum cuts for arbitrary choice of

*edge*weights. ...

*In*particular, we thank a referee who has suggested the

*algorithmic*variant for the method of [45] outlined

*in*the end of Section 5. ...

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