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Fitting Planar Graphs on Planar Maps

2015
*
Journal of Graph Algorithms and Applications
*

We consider a problem that combines aspects of both by studying the problem of fitting

doi:10.7155/jgaa.00367
fatcat:ngdvntnbyraaljyp3phm5xbi54
*planar**graphs*on*planar**maps*. ...*Graph*and cartographic visualization have the common objective to provide intuitive understanding of some underlying data. ... Acknowledgments We thank the two anonymous referees for useful suggestion and for providing us with sketch about fitting*graphs*with a bend per each inter-cluster edge. ...##
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Fitting Planar Graphs on Planar Maps
[chapter]

2014
*
Lecture Notes in Computer Science
*

We consider a problem that combines aspects of both by studying the problem of fitting

doi:10.1007/978-3-319-04298-5_6
fatcat:ipcr3mskk5fjppqat2lbqmkh5a
*planar**graphs*on*planar**maps*. ...*Graph*and cartographic visualization have the common objective to provide intuitive understanding of some underlying data. ... Acknowledgments We thank the two anonymous referees for useful suggestion and for providing us with sketch about fitting*graphs*with a bend per each inter-cluster edge. ...##
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Mapping planar graphs into the Coxeter graph

2016
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Discrete Mathematics
*

We conjecture that every

doi:10.1016/j.disc.2015.10.007
fatcat:w3gpf4nhbjfrneobtdzqb44tw4
*planar**graph*of odd-girth at least 11 admits a homomorphism to the Coxeter*graph*. ... Supporting this conjecture, we prove that every*planar**graph*of odd-girth at least 17 admits a homomorphism to the Coxeter*graph*. ... A plane*graph*is a*planar**graph*with a given*planar*embedding. A homomorphism of a*graph*G to another*graph*H is a*mapping*ϕ : V (G) → V (H) which preserves adjacency. ...##
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Enumeration of chordal planar graphs and maps
[article]

2022
*
arXiv
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pre-print

Chordal

arXiv:2202.13340v2
fatcat:plxgcjq6erhknktg65a5ybrnhu
*planar**graphs*(or*maps*) are a natural example of a subcritical class of*graphs*in which the class of 3-connected*graphs*is relatively rich. ... We also determine the number of rooted simple chordal*planar**maps*with n edges, which is asymptotically c_2 n^-3/2δ^n, where δ = 1/σ≈ 6.40375, and σ is an algebraic number of degree 12. ... Simple chordal*planar**maps*Decomposition of 2-connected simple chordal*maps*. ...##
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The monadic second-order logic of graphs XII: planar graphs and planar maps

2000
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Theoretical Computer Science
*

The

doi:10.1016/s0304-3975(99)00305-9
fatcat:mtgylluzfzb5ve45ngvzgarcpe
*planar*embedding of a*graph*can be speciÿed by a relational structure called a*map*, which is a*graph*enriched with a circular ordering of the edges incident with each vertex. ... Hence, we have for*planar**graphs*, some kind of equivalence between linear orderings and*planar*embeddings. ... There exists an MS-transduction that transforms an arbitrary*map*of a*planar**graph*into a*planar**map*of the same*graph*. ...##
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A mating-of-trees approach for graph distances in random planar maps
[article]

2020
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arXiv
*
pre-print

Using our technique, we prove estimates for

arXiv:1711.00723v3
fatcat:3kxhmoiagzc35h6iyr45vigaka
*graph*distances in the above family of random*planar**maps*. ... The basic idea of our approach is to compare a given random*planar**map*M to a mated-CRT*map*—a random*planar**map*constructed from a correlated two-dimensional Brownian motion—using a strong coupling (Zaitsev ... define a*planar**map*to be a*graph*embedded in the 2-sphere, rather than in the plane. ...##
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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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Short encodings of planar graphs and maps

1995
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Discrete Applied Mathematics
*

We discuss space-efficient encoding schemes for

doi:10.1016/0166-218x(93)e0150-w
fatcat:ergpesoz3zcfhcdrshey2z7kui
*planar**graphs*and*maps*. Our results improve on the constants of previous schemes and can be achieved with simple encoding algorithms. ... Since every*map*embeds a unique*planar**graph*, this scheme also suffices to encode loop-and stick-free*planar**graphs*. ... In encoding a*graph*we must encode the incidences among vertexes and edges. By*maps*we understand topological equivalence classes of*planar*embeddings of*planar**graphs*. ...##
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Random planar maps and graphs with minimum degree two and three
[article]

2018
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arXiv
*
pre-print

We find precise asymptotic estimates for the number of

arXiv:1403.5211v2
fatcat:ymuyxrzpgrbn5k5n2fszgfsvau
*planar**maps*and*graphs*with a condition on the minimum degree, and properties of random*graphs*from these classes. ... These results provide new information on the structure of random*planar**graphs*. ...*Planar**graphs*In this section we follow the ideas of Section 3 on*planar**maps*in order to obtain related results for*planar*2-*graphs*and 3-*graphs*. ...##
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On the Relationship Between Map Graphs and Clique Planar Graphs
[chapter]

2015
*
Lecture Notes in Computer Science
*

Figure 2 provides an example of a

doi:10.1007/978-3-319-27261-0_46
fatcat:hkdn2tj63jemdj7kbaa6tv5w7y
*graph*that is both a*map**graph*and a clique*planar**graph*. ... There exists a clique*planar**graph*that is not a*map**graph*. Proof. Consider the*graph*G of Fig. 2(a) . ... A*graph*(a) that is both a*map**graph*(b) and a clique*planar**graph*(c) A*graph*(a) that is clique*planar*(b) but not a*map**graph*(c) (Color figure online). ...##
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Nodally 3-connected planar graphs and convex combination mappings
[article]

2007
*
arXiv
*
pre-print

If a

arXiv:0708.0964v1
fatcat:pgk7zry7afe67f2klr72hr6k6m
*planar**graph*is nodally 3-connected or triangulated then every such*mapping*is an embedding (Tutte, Floater). ... A convex combination*mapping*of a*planar**graph*is a plane*mapping*in which the external vertices are*mapped*to the corners of a convex polygon and every internal vertex is a proper weighted average of ... The*graph*in Figure 4 is nodally 3-connected but not triconnected. A fully triangulated*planar**graph*is a triangulated*planar**graph*in which there are three external edges. ...##
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Planar Graphs, via Well-Orderly Maps and Trees

2006
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Graphs and Combinatorics
*

The family of well-orderly

doi:10.1007/s00373-006-0647-2
fatcat:er2iupjnnnb2hbtlsjeggjt3fi
*maps*is a family of*planar**maps*with the property that every connected*planar**graph*has at least one plane embedding which is a well-orderly*map*. ... A direct consequence of this is a new upper bound on the number p(n) of unlabeled*planar**graphs*with n nodes, log 2 p(n) 4.91n. ...*Planar**Graphs*and Well-Orderly*Maps*A*planar**map*(or plane*graph*) is an embedding of a connected*planar**graph*on the plane so that edges meet only at their endpoints. ...##
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Topological maps based on graphs of planar regions

2009
*
2009 IEEE/RSJ International Conference on Intelligent Robots and Systems
*

First the triple matching approach to extract all the planes seen in the set of images and second a new topological

doi:10.1109/iros.2009.5354184
dblp:conf/iros/MontijanoS09
fatcat:5sigwpab2rasdn5rqse7ls2pdi
*map*construction based on a*graph*of*planar*regions which can be used by mobile robots ... We propose here a new hierarchical structure in which landmarks extracted from conventional images are grouped creating a*graph*of*planar*regions. ... Index Terms -homography, topological*maps*,*planar*regions. I. ...##
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Planar Graphs, via Well-Orderly Maps and Trees
[chapter]

2004
*
Lecture Notes in Computer Science
*

The family of well-orderly

doi:10.1007/978-3-540-30559-0_23
fatcat:hwqgggbsibabvfbwjqqmpfiaf4
*maps*is a family of*planar**maps*with the property that every connected*planar**graph*has at least one plane embedding which is a well-orderly*map*. ... A direct consequence of this is a new upper bound on the number p(n) of unlabeled*planar**graphs*with n nodes, log 2 p(n) 4.91n. ...*Planar**Graphs*and Well-Orderly*Maps*A*planar**map*(or plane*graph*) is an embedding of a connected*planar**graph*on the plane so that edges meet only at their endpoints. ...##
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Unavoidable set of face types for planar maps

1996
*
Discussiones Mathematicae Graph Theory
*

The type of a face f of a

doi:10.7151/dmgt.1028
fatcat:ayneet3puzf73kj4q44fmstp2i
*planar**map*is a sequence of degrees of vertices of f as they are encountered when traversing the boundary of f . ... Edges of small weights in*planar**maps*of minimum degree 5 are investigated in a very recent paper by Borodin and Sanders [9]. ... Let G n m (P ) be a*planar**graph*with V (G n m (P )) = {x i : i ∈ [1, mn]} ∪ {y 0 , y 1 }, ∞ i=3 b − i (T ). ...
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