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Fitting Planar Graphs on Planar Maps

Md. Jawaherul Alam, Michael Kaufmann, Stephen G. Kobourov, Tamara Mchedlidze
2015 Journal of Graph Algorithms and Applications  
We consider a problem that combines aspects of both by studying the problem of fitting 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.  ... 
doi:10.7155/jgaa.00367 fatcat:ngdvntnbyraaljyp3phm5xbi54

Fitting Planar Graphs on Planar Maps [chapter]

Md. Jawaherul Alam, Michael Kaufmann, Stephen G. Kobourov, Tamara Mchedlidze
2014 Lecture Notes in Computer Science  
We consider a problem that combines aspects of both by studying the problem of fitting 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.  ... 
doi:10.1007/978-3-319-04298-5_6 fatcat:ipcr3mskk5fjppqat2lbqmkh5a

Mapping planar graphs into the Coxeter graph

Ararat Harutyunyan, Reza Naserasr, Mirko Petruševski, Riste Škrekovski, Qiang Sun
2016 Discrete Mathematics  
We conjecture that every 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.  ... 
doi:10.1016/j.disc.2015.10.007 fatcat:w3gpf4nhbjfrneobtdzqb44tw4

Enumeration of chordal planar graphs and maps [article]

Jordi Castellví, Marc Noy, Clément Requilé
2022 arXiv   pre-print
Chordal 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.  ... 
arXiv:2202.13340v2 fatcat:plxgcjq6erhknktg65a5ybrnhu

The monadic second-order logic of graphs XII: planar graphs and planar maps

Bruno Courcelle
2000 Theoretical Computer Science  
The 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.  ... 
doi:10.1016/s0304-3975(99)00305-9 fatcat:mtgylluzfzb5ve45ngvzgarcpe

A mating-of-trees approach for graph distances in random planar maps [article]

Ewain Gwynne, Nina Holden, Xin Sun
2020 arXiv   pre-print
Using our technique, we prove estimates for 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.  ... 
arXiv:1711.00723v3 fatcat:3kxhmoiagzc35h6iyr45vigaka

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

Short encodings of planar graphs and maps

Kenneth Keeler, Jeffery Westbrook
1995 Discrete Applied Mathematics  
We discuss space-efficient encoding schemes for 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.  ... 
doi:10.1016/0166-218x(93)e0150-w fatcat:ergpesoz3zcfhcdrshey2z7kui

Random planar maps and graphs with minimum degree two and three [article]

Marc Noy, Lander Ramos
2018 arXiv   pre-print
We find precise asymptotic estimates for the number of 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.  ... 
arXiv:1403.5211v2 fatcat:ymuyxrzpgrbn5k5n2fszgfsvau

On the Relationship Between Map Graphs and Clique Planar Graphs [chapter]

Patrizio Angelini, Giordano Da Lozzo, Giuseppe Di Battista, Fabrizio Frati, Maurizio Patrignani, Ignaz Rutter
2015 Lecture Notes in Computer Science  
Figure 2 provides an example of a 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).  ... 
doi:10.1007/978-3-319-27261-0_46 fatcat:hkdn2tj63jemdj7kbaa6tv5w7y

Nodally 3-connected planar graphs and convex combination mappings [article]

Colm O Dunlaing
2007 arXiv   pre-print
If a 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.  ... 
arXiv:0708.0964v1 fatcat:pgk7zry7afe67f2klr72hr6k6m

Planar Graphs, via Well-Orderly Maps and Trees

Nicolas Bonichon, Cyril Gavoille, Nicolas Hanusse, Dominique Poulalhon, Gilles Schaeffer
2006 Graphs and Combinatorics  
The family of well-orderly 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.  ... 
doi:10.1007/s00373-006-0647-2 fatcat:er2iupjnnnb2hbtlsjeggjt3fi

Topological maps based on graphs of planar regions

E. Montijano, C. Sagues
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 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.  ... 
doi:10.1109/iros.2009.5354184 dblp:conf/iros/MontijanoS09 fatcat:5sigwpab2rasdn5rqse7ls2pdi

Planar Graphs, via Well-Orderly Maps and Trees [chapter]

Nicolas Bonichon, Cyril Gavoille, Nicolas Hanusse, Dominique Poulalhon, Gilles Schaeffer
2004 Lecture Notes in Computer Science  
The family of well-orderly 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.  ... 
doi:10.1007/978-3-540-30559-0_23 fatcat:hwqgggbsibabvfbwjqqmpfiaf4

Unavoidable set of face types for planar maps

Mirko Horňák, Stanislav Jendrol'
1996 Discussiones Mathematicae Graph Theory  
The type of a face f of a 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 ).  ... 
doi:10.7151/dmgt.1028 fatcat:ayneet3puzf73kj4q44fmstp2i
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