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

Henry Förster, Michael Kaufmann, Chrysanthi N. Raftopoulou
2021 arXiv   pre-print
In this paper, we consider the recognition of 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 }.  ... 
arXiv:2108.00665v2 fatcat:vv7lfqujyzfrvp5wb7xu7p4zmq

On 4-Map Graphs and 1-Planar Graphs and their Recognition Problem [article]

Franz J. Brandenburg
2015 arXiv   pre-print
We establish a one-to-one correspondence between 1-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.  ... 
arXiv:1509.03447v1 fatcat:daiz4wbyyrardlfvfsan2jetgq

NIC-Planar Graphs [article]

Christian Bachmaier, Franz J. Brandenburg, Kathrin Hanauer, Daniel Neuwirth, Josef Reislhuber
2017 arXiv   pre-print
Further, we prove that 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.  ... 
arXiv:1701.04375v3 fatcat:gowtlgenuzh4hpztqkd4pnfiha

An annotated bibliography on 1-planarity

Stephen G. Kobourov, Giuseppe Liotta, Fabrizio Montecchiani
2017 Computer Science Review  
The notion of 1-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.  ... 
doi:10.1016/j.cosrev.2017.06.002 fatcat:sjx2s5s4a5awzpsylfoc5upfmi

A Survey on Graph Drawing Beyond Planarity [article]

Walter Didimo, Giuseppe Liotta, Fabrizio Montecchiani
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.  ... 
arXiv:1804.07257v1 fatcat:xeckxilhfbbgdbfhctikymlq6m

Recognizing Optimal 1-Planar Graphs in Linear Time

Franz J. Brandenburg
2016 Algorithmica  
We show that 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.  ... 
doi:10.1007/s00453-016-0226-8 fatcat:us4pwhvfffa4rfw7cwlkxijrhq

An external-memory depth-first search algorithm for general grid graphs

Jun-Ho Her, R.S. Ramakrishna
2007 Theoretical Computer Science  
The I/O-complexity of the algorithm is O(sort(N ) log 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.  ... 
doi:10.1016/j.tcs.2006.12.022 fatcat:wnhms3ejwfcizbqpq4m7iscdqy

Recognizing IC-Planar and NIC-Planar Graphs [article]

Franz J. Brandenburg
2016 arXiv   pre-print
In consequence, 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] .  ... 
arXiv:1610.08884v1 fatcat:57ujl6sbpjazhhtprxcesfabxm

A Reduction System for Optimal 1-Planar Graphs [article]

Franz J. Brandenburg
2016 arXiv   pre-print
A 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  ... 
arXiv:1602.06407v2 fatcat:rqjf7brxvvg3jhbavm52dsqwky

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.  ... 

Beyond-Planar Graphs: Algorithmics and Combinatorics (Dagstuhl Seminar 16452)

Sok-Hee Hong, Michael Kaufmann, Stephen G. Kobourov, János Pach, Marc Herbstritt
2017 Dagstuhl Reports  
This report summarizes Dagstuhl Seminar 16452 "Beyond-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 ∆?  ... 
doi:10.4230/dagrep.6.11.35 dblp:journals/dagstuhl-reports/Hong0KP16 fatcat:gmft5jywxbapza52lhj4o23ngq

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.  ... 

1-Bend RAC Drawings of 1-Planar Graphs [article]

Walter Didimo, Giuseppe Liotta, Saeed Mehrabi, Fabrizio Montecchiani
2016 arXiv   pre-print
The relationships between 1-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.  ... 
arXiv:1608.08418v1 fatcat:rzcne3zy2zbk3kbscfapxjdew4

An Experimental Study of a 1-planarity Testing and Embedding Algorithm [article]

Carla Binucci, Walter Didimo, Fabrizio Montecchiani
2019 arXiv   pre-print
We investigate the feasibility of a 1-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  ... 
arXiv:1911.00573v1 fatcat:qf6pum4rvnbtpjymtrh4nwh4cy

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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