Planar Maps

2015 ◽  
pp. 335-395
Author(s):  
Gilles Schaeffer
Keyword(s):  
Planar Maps

THE DYNAMICS OF NQ-SYSTEMS IN THE PLANE

2009 ◽  
Vol 19 (01) ◽  
pp. 117-133 ◽  
Author(s):  
MATEJ MENCINGER ◽  
MILAN KUTNJAK
Keyword(s):  
Algebraic Approach ◽  
Dynamical Analysis ◽  
Algebraic Classification ◽  
Commutative Algebras ◽  
Planar Maps ◽  
Points Of View ◽  
One To One

The dynamics of discrete homogeneous quadratic planar maps is considered via the algebraic approach. There is a one-to-one correspondence between these systems and 2D commutative algebras (c.f. [Markus, 1960]). In particular, we consider the systems corresponding to algebras which contain some nilpotents of rank two (i.e. NQ-systems). Markus algebraic classification is used to obtain the class representatives. The case-by-case dynamical analysis is presented. It is proven that there is no chaos in NQ-systems. Yet, some cases are really interesting from the dynamical and bifurcational points of view.

The Design and Implementation of Planar Maps in CGAL

1999 ◽  
pp. 154-168 ◽  
Author(s):  
Eyal Flato ◽  
Dan Halperin ◽  
Iddo Hanniel ◽  
Oren Nechushtan
Keyword(s):  
Design And Implementation ◽  
Planar Maps

scholarly journals Towards a classification of planar maps

2018 ◽  
Vol 98 (6) ◽  
Author(s):  
Alexandre Diet ◽  
Marc Barthelemy
Keyword(s):  
Planar Maps

scholarly journals Rooted planar maps modulo some patterns

2016 ◽  
Vol 339 (4) ◽  
pp. 1199-1205
Author(s):  
Jean-Luc Baril ◽  
Richard Genestier ◽  
Alain Giorgetti ◽  
Armen Petrossian
Keyword(s):  
Planar Maps

Planar Maps

1991 ◽  
pp. 443-494
Author(s):  
Jack K. Hale ◽  
Hüseyin Koçak
Keyword(s):  
Planar Maps

scholarly journals The boundary of random planar maps via looptrees

2020 ◽  
Vol 29 (2) ◽  
pp. 391-430
Author(s):  
Igor Kortchemski ◽  
Loïc Richier
Keyword(s):  
Planar Maps

scholarly journals Global dynamics for symmetric planar maps

2013 ◽  
Vol 33 (6) ◽  
pp. 2241-2251 ◽  
Author(s):  
Begoña Alarcón ◽  
◽  
Sofia B. S. D. Castro ◽  
Isabel S. Labouriau ◽  
◽  
...  
Keyword(s):  
Global Dynamics ◽  
Planar Maps

scholarly journals On the Riemann surface type of random planar maps

2013 ◽  
Vol 29 (3) ◽  
pp. 1071-1090 ◽  
Author(s):  
James Gill ◽  
Steffen Rohde
Keyword(s):  
Riemann Surface ◽  
Surface Type ◽  
Planar Maps

scholarly journals A Generic Method for Bijections between Blossoming Trees and Planar Maps

10.37236/3386 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Marie Albenque ◽  
Dominique Poulalhon
Keyword(s):  
Recent Work ◽  
Spanning Tree ◽  
Linear Time ◽  
Computational Scheme ◽  
Closed Formula ◽  
Special Cases ◽  
Planar Maps ◽  
Root Face ◽  
By Products

This article presents a unified bijective scheme between planar maps and blossoming trees, where a blossoming tree is defined as a spanning tree of the map decorated with some dangling half-edges that enable to reconstruct its faces. Our method generalizes a previous construction of Bernardi by loosening its conditions of application so as to include annular maps, that is maps embedded in the plane with a root face different from the outer face.The bijective construction presented here relies deeply on the theory of $\alpha$-orientations introduced by Felsner, and in particular on the existence of minimal and accessible orientations. Since most of the families of maps can be characterized by such orientations, our generic bijective method is proved to capture as special cases many previously known bijections involving blossoming trees: for example Eulerian maps, $m$-Eulerian maps, non-separable maps and simple triangulations and quadrangulations of a $k$-gon. Moreover, it also permits to obtain new bijective constructions for bipolar orientations and $d$-angulations of girth $d$ of a $k$-gon.As for applications, each specialization of the construction translates into enumerative by-products, either via a closed formula or via a recursive computational scheme. Besides, for every family of maps described in the paper, the construction can be implemented in linear time. It yields thus an effective way to encode or sample planar maps.In a recent work, Bernardi and Fusy introduced another unified bijective scheme; we adopt here a different strategy which allows us to capture different bijections. These two approaches should be seen as two complementary ways of unifying bijections between planar maps and decorated trees.

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