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Recognizing and Embedding Simple Optimal 2-Planar Graphs
[article]

2021
*
arXiv
*
pre-print

In this paper, we consider the recognition of

arXiv:2108.00665v2
fatcat:vv7lfqujyzfrvp5wb7xu7p4zmq
*simple**optimal**2*-*planar**graphs*. ... We exploit a combinatorial characterization of such*graphs**and*present a linear time algorithm for recognition*and**embedding*. ...*Recognizing**and**Embedding**Simple**Optimal**2*-*Planar**Graphs*11 Lemma 4.11 (∗). Let G be*optimal**2*-*planar**and*let T = {T }. ...##
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On 4-Map Graphs and 1-Planar Graphs and their Recognition Problem
[article]

2015
*
arXiv
*
pre-print

We establish a one-to-one correspondence between 1-

arXiv:1509.03447v1
fatcat:daiz4wbyyrardlfvfsan2jetgq
*planar**graphs**and*general*and*hole-free 4-map*graphs**and*show that 1-*planar**graphs*can be*recognized*in polynomial time if they are crossing-augmented ... , fully triangulated,*and*maximal 1-*planar*, respectively, with a polynomial of degree 120, 3,*and*5, respectively. ... Acknowledgements I would like to thank Christian Bachmaier for many inspiring discussions*and*his support,*and*Giuseppe Liotta for the hint on map*graphs*. ...##
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NIC-Planar Graphs
[article]

2017
*
arXiv
*
pre-print

Further, we prove that

arXiv:1701.04375v3
fatcat:gowtlgenuzh4hpztqkd4pnfiha
*optimal*NIC-*planar**graphs*with 3.6(n-*2*) edges have a unique*embedding**and*can be*recognized*in linear time, whereas the general recognition problem of NIC-*planar**graphs*is NP-complete ... The characterization is used to derive tight bounds on the density of maximal NIC-*planar**graphs*which ranges between 3.2(n-*2*)*and*3.6(n-*2*). ...*Recognizing**Optimal*NIC-*Planar**Graphs*in Linear Time The study of maximal NIC-*planar**graphs*in Sect. 3 also provides a key to a linear-time recognition algorithm for*optimal*NIC-*planar**graphs*. ...##
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An annotated bibliography on 1-planarity

2017
*
Computer Science Review
*

The notion of 1-

doi:10.1016/j.cosrev.2017.06.002
fatcat:sjx2s5s4a5awzpsylfoc5upfmi
*planarity*is among the most natural*and*most studied generalizations of*graph**planarity*. A*graph*is 1-*planar*if it has an*embedding*where each edge is crossed by at most another edge. ... The study of 1-*planar**graphs*dates back to more than fifty years ago*and*, recently, it has driven increasing attention in the areas of*graph*theory,*graph*algorithms,*graph*drawing,*and*computational geometry ... Bekos, Franz Brandenburg, Walter Didimo, David Eppstein, Michael Kaufmann, Sergey Pupyrev,*and*David R. Wood for great discussions, explanations,*and*references. ...##
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A Survey on Graph Drawing Beyond Planarity
[article]

2018
*
arXiv
*
pre-print

*Graph*Drawing Beyond

*Planarity*is a rapidly growing research area that classifies

*and*studies geometric representations of non-

*planar*

*graphs*in terms of forbidden crossing configurations. ... Aim of this survey is to describe the main research directions in this area, the most prominent known results,

*and*some of the most challenging open problems. ... In particular, the resulting bounds are tight for k = 1

*and*for k =

*2*, which means that the

*optimal*1-

*planar*

*graphs*have 4n − 8 edges

*and*the

*optimal*

*2*-

*planar*

*graphs*have 5n − 10 edges. ...

##
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Recognizing Optimal 1-Planar Graphs in Linear Time

2016
*
Algorithmica
*

We show that

doi:10.1007/s00453-016-0226-8
fatcat:us4pwhvfffa4rfw7cwlkxijrhq
*optimal*1-*planar**graphs*can be*recognized*in linear time. ... A*graph*with n vertices is 1-*planar*if it can be drawn in the plane such that each edge is crossed at most once,*and*is*optimal*if it has the maximum of 4n-8 edges. ... Acknowledgements I would like to thank Christian Bachmaier*and*Josef Reislhuber for many inspiring discussions*and*their support*and*the reviewers for their valuable suggestions. ...##
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An external-memory depth-first search algorithm for general grid graphs

2007
*
Theoretical Computer Science
*

The I/O-complexity of the algorithm is O(sort(N ) log

doi:10.1016/j.tcs.2006.12.022
fatcat:wnhms3ejwfcizbqpq4m7iscdqy
*2*N M ), where N = |V | + |E|, sort(N ) = Θ( N B log M/B N B ) is the sorting I/O-complexity, M is the memory size,*and*B is the block size. ...*Graph*data in modern scientific*and*engineering applications are often too large to fit in the computer's main memory. Input/output (I/O) complexity is a major research issue in this context. ... cycle*2*3 separator of the*planar*subgraph after Step 3 plays the role of a*simple*cycle*2*3 part*recognizer*in the given biconnected grid*graph*. ...##
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Recognizing IC-Planar and NIC-Planar Graphs
[article]

2016
*
arXiv
*
pre-print

In consequence,

arXiv:1610.08884v1
fatcat:57ujl6sbpjazhhtprxcesfabxm
*planar*-maximal*and*maximal IC-*planar**and*NIC-*planar**graphs*can be*recognized*in O(n^5) time*and*maximum*and**optimal*ones in O(n^3) time. ... We prove that triangulated IC-*planar**and*NIC-*planar**graphs*can be*recognized*in cubic time. A*graph*is 1-*planar*if it can be drawn in the plane with at most one crossing per edge. ... There are*optimal*IC-*planar**graphs*only for n = 4k*and**optimal*NIC-*planar**graphs*only for n = 5t +*2**and*such*graphs*exist for all k ≥*2*[38]*and*all t ≥*2*[5] . ...##
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A Reduction System for Optimal 1-Planar Graphs
[article]

2016
*
arXiv
*
pre-print

A

arXiv:1602.06407v2
fatcat:rqjf7brxvvg3jhbavm52dsqwky
*graph*is*optimal*1-*planar*if it can be drawn in the plane with at most one crossing per edge*and*is*optimal*if it has the maximum of 4n-8 edges. ... There is a*graph*reduction system so that every*optimal*1-*planar**graph*can be reduced to an irreducible extended wheel*graph*, provided the reductions are applied such that the given*graph*class is preserved ... We use the fact that*optimal*1-*planar**graphs*have a unique*embedding*(except for extended wheel*graphs*) [27, 28]*and*identify a*graph**and*its 1-*planar**embedding*as well as the*graphs*of CS*and*CC*and*...##
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Page 7073 of Mathematical Reviews Vol. , Issue 2003i
[page]

2003
*
Mathematical Reviews
*

Second, we give a polyno- mial time algorithm for computing ‘

*optimal*’ quasi-upward*planar*drawings within a given*planar**embedding*. ... In this paper we characterize the set of all upward*embeddings**and*orientations of a plane*graph*by using a*simple*flow model. ...##
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Beyond-Planar Graphs: Algorithmics and Combinatorics (Dagstuhl Seminar 16452)

2017
*
Dagstuhl Reports
*

This report summarizes Dagstuhl Seminar 16452 "Beyond-

doi:10.4230/dagrep.6.11.35
dblp:journals/dagstuhl-reports/Hong0KP16
fatcat:gmft5jywxbapza52lhj4o23ngq
*Planar**Graphs*: Algorithmics*and*Combinatorics"*and*documents the talks*and*discussions. ... 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 ... What is the complexity of*recognizing*outer-quasi-*planar*or*2*-layer quasiplanar*graphs*with maximum degree ∆? ...##
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Page 5114 of Mathematical Reviews Vol. , Issue 911
[page]

1991
*
Mathematical Reviews
*

Our results use a new result on

*embedding**graphs*in a single-layer grid, namely an*embedding*of n-node*planar**graphs*such that each edge makes at most four turns,*and*all nodes are*embedded*on the same ... We also show that*planar**graphs*of bounded degree can be*embedded*on two layers in O(n>/?(log n)) area. ...##
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1-Bend RAC Drawings of 1-Planar Graphs
[article]

2016
*
arXiv
*
pre-print

The relationships between 1-

arXiv:1608.08418v1
fatcat:rzcne3zy2zbk3kbscfapxjdew4
*planar**graphs**and*RAC drawings have been partially studied in the literature. ... It is known that there are both 1-*planar**graphs*that are not straight-line RAC drawable*and**graphs*that have a straight-line RAC drawing but that are not 1-*planar*. ... This topic is informally*recognized*as "beyond*planarity*". Different types of properties give rise to different families of beyond*planar**graphs*. ...##
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An Experimental Study of a 1-planarity Testing and Embedding Algorithm
[article]

2019
*
arXiv
*
pre-print

We investigate the feasibility of a 1-

arXiv:1911.00573v1
fatcat:qf6pum4rvnbtpjymtrh4nwh4cy
*planarity*testing*and**embedding*algorithm based on a backtracking strategy. ... Although several polynomial-time algorithms have been described for*recognizing*specific subfamilies of 1-*planar**graphs*, no implementations of general algorithms are available to date. ... Namely, let E be the set of all (unordered) pairs of edges that can cross in some*embedding*of G, i.e., the pairs {e 1 , e*2*} such that e 1 ∈ E*and*e*2*∈ E are independent (we only consider*simple*drawings ...##
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Page 1261 of Mathematical Reviews Vol. , Issue 92c
[page]

1992
*
Mathematical Reviews
*

It is shown there are trivalent

*planar**graphs*of order n for which any*2*-page*embedding*has pagewidth Q(n). A. G. ... This leads to efficient algorithms for*recognizing*chordal comparability*graphs**and*trapezoid*graphs*, computing the transitive closure of two-dimensional partial orders,*and*a number of other problems. ...
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