scholarly journals Fitting Planar Graphs on Planar Maps

2014 ◽  
pp. 52-64
Author(s):  
Md. Jawaherul Alam ◽  
Michael Kaufmann ◽  
Stephen G. Kobourov ◽  
Tamara Mchedlidze
Keyword(s):  
Planar Graphs ◽  
Planar Maps

scholarly journals Fitting Planar Graphs on Planar Maps

2015 ◽  
Vol 19 (1) ◽  
pp. 413-440 ◽  
Author(s):  
Md. Jawaherul Alam ◽  
Michael Kaufmann ◽  
Stephen G. Kobourov ◽  
Tamara Mchedlidze
Keyword(s):  
Planar Graphs ◽  
Planar Maps

scholarly journals A Complete Grammar for Decomposing a Family of Graphs into 3-Connected Components

10.37236/872 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Guillaume Chapuy ◽  
Éric Fusy ◽  
Mihyun Kang ◽  
Bilyana Shoilekova
Keyword(s):  
Planar Graphs ◽  
Equation System ◽  
Canonical Decomposition ◽  
Connected Components ◽  
Asymptotic Enumeration ◽  
Important Ingredient ◽  
Limit Laws ◽  
Promising Tool ◽  
Analytic Expression ◽  
Planar Maps

Tutte has described in the book "Connectivity in graphs" a canonical decomposition of any graph into 3-connected components. In this article we translate (using the language of symbolic combinatorics) Tutte's decomposition into a general grammar expressing any family ${\cal G}$ of graphs (with some stability conditions) in terms of the subfamily ${\cal G}_3$ of graphs in ${\cal G}$ that are 3-connected (until now, such a general grammar was only known for the decomposition into $2$-connected components). As a byproduct, our grammar yields an explicit system of equations to express the series counting a (labelled) family of graphs in terms of the series counting the subfamily of $3$-connected graphs. A key ingredient we use is an extension of the so-called dissymmetry theorem, which yields negative signs in the grammar and associated equation system, but has the considerable advantage of avoiding the difficult integration steps that appear with other approaches, in particular in recent work by Giménez and Noy on counting planar graphs. As a main application we recover in a purely combinatorial way the analytic expression found by Giménez and Noy for the series counting labelled planar graphs (such an expression is crucial to do asymptotic enumeration and to obtain limit laws of various parameters on random planar graphs). Besides the grammar, an important ingredient of our method is a recent bijective construction of planar maps by Bouttier, Di Francesco and Guitter. Finally, our grammar applies also to the case of unlabelled structures, since the dissymetry theorem takes symmetries into account. Even if there are still difficulties in counting unlabelled 3-connected planar graphs, we think that our grammar is a promising tool toward the asymptotic enumeration of unlabelled planar graphs, since it circumvents some difficult integral calculations.

scholarly journals Random Planar Maps and Graphs with Minimum Degree Two and Three

10.37236/7640 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Marc Noy ◽  
Lander Ramos
Keyword(s):  
Planar Graph ◽  
Random Graphs ◽  
Planar Graphs ◽  
Minimum Degree ◽  
Asymptotic Estimates ◽  
The Core ◽  
New Information ◽  
Planar Maps ◽  
Explicit Constant ◽  
Precise Asymptotic

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. In particular we show that the size of the largest tree attached to the core of a random planar graph is of order $c \log(n)$ for an explicit constant $c$. These results provide new information on the structure of random planar graphs.

Acute Constraints in Straight-Line Drawings of Planar Graphs

2019 ◽  
Vol E102.A (9) ◽  
pp. 994-1001
Author(s):  
Akane SETO ◽  
Aleksandar SHURBEVSKI ◽  
Hiroshi NAGAMOCHI ◽  
Peter EADES
Keyword(s):  
Planar Graphs ◽  
Line Drawings ◽  
Straight Line

A Space-Efficient Separator Algorithm for Planar Graphs

2019 ◽  
Vol E102.A (9) ◽  
pp. 1007-1016 ◽  
Author(s):  
Ryo ASHIDA ◽  
Sebastian KUHNERT ◽  
Osamu WATANABE
Keyword(s):  
Planar Graphs

Note on (semi-)proper orientation of some triangulated planar graphs

2021 ◽  
Vol 392 ◽  
pp. 125723
Author(s):  
Ruijuan Gu ◽  
Hui Lei ◽  
Yulai Ma ◽  
Zhenyu Taoqiu
Keyword(s):  
Planar Graphs ◽  
Proper Orientation

scholarly journals Distributed Dominating Set Approximations beyond Planar Graphs

2019 ◽  
Vol 15 (3) ◽  
pp. 1-18 ◽  
Author(s):  
Saeed Akhoondian Amiri ◽  
Stefan Schmid ◽  
Sebastian Siebertz
Keyword(s):  
Planar Graphs ◽  
Dominating Set

scholarly journals Clustered 3-colouring graphs of bounded degree

2021 ◽  
pp. 1-13
Author(s):  
Vida Dujmović ◽  
Louis Esperet ◽  
Pat Morin ◽  
Bartosz Walczak ◽  
David R. Wood
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Bounded Degree ◽  
Bounded Number ◽  
Proper Vertex ◽  
Vertex Colouring

Abstract A (not necessarily proper) vertex colouring of a graph has clustering c if every monochromatic component has at most c vertices. We prove that planar graphs with maximum degree $\Delta$ are 3-colourable with clustering $O(\Delta^2)$ . The previous best bound was $O(\Delta^{37})$ . This result for planar graphs generalises to graphs that can be drawn on a surface of bounded Euler genus with a bounded number of crossings per edge. We then prove that graphs with maximum degree $\Delta$ that exclude a fixed minor are 3-colourable with clustering $O(\Delta^5)$ . The best previous bound for this result was exponential in $\Delta$ .

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