scholarly journals Boundaries of planar graphs, via circle packings

2016 ◽  
Vol 44 (3) ◽  
pp. 1956-1984 ◽  
Author(s):  
Omer Angel ◽  
Martin T. Barlow ◽  
Ori Gurel-Gurevich ◽  
Asaf Nachmias
Keyword(s):  
Planar Graphs ◽  
Circle Packings

scholarly journals Circle packings, kissing reflection groups and critically fixed anti-rational maps

2022 ◽  
Vol 10 ◽  
Author(s):  
Russell Lodge ◽  
Yusheng Luo ◽  
Sabyasachi Mukherjee
Keyword(s):  
Planar Graphs ◽  
Rational Maps ◽  
Reflection Groups ◽  
Circle Packings

Abstract In this article, we establish an explicit correspondence between kissing reflection groups and critically fixed anti-rational maps. The correspondence, which is expressed using simple planar graphs, has several dynamical consequences. As an application of this correspondence, we give complete answers to geometric mating problems for critically fixed anti-rational maps.

Harmonic functions on planar and almost planar graphs and manifolds, via circle packings

1996 ◽  
Vol 126 (3) ◽  
pp. 565-587 ◽  
Author(s):  
Itai Benjamini ◽  
Oded Schramm
Keyword(s):  
Harmonic Functions ◽  
Planar Graphs ◽  
Circle Packings

scholarly journals Balanced Circle Packings for Planar Graphs

2014 ◽  
pp. 125-136 ◽  
Author(s):  
Md. Jawaherul Alam ◽  
David Eppstein ◽  
Michael T. Goodrich ◽  
Stephen G. Kobourov ◽  
Sergey Pupyrev
Keyword(s):  
Planar Graphs ◽  
Circle Packings

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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