Gromov hyperbolicity of planar graphs

We prove that under appropriate assumptions adding or removing an infinite amount of edges to a given planar graph preserves its non-hyperbolicity, a result which is shown to be false in general. In particular, we make a conjecture that every tessellation graph of ℝ 2 with convex tiles is non-hyperb...

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Bibliographic Details
Published in:Central European journal of mathematics 2013, Vol.11 (10), p.1817-1830
Main Authors: Cantón, Alicia, Granados, Ana, Pestana, Domingo, Rodríguez, José M.
Format: Article
Language:English
Subjects:
05A20
05C10
05C63
05C75
Algebra
Geodesics
Geometry
Graph theory
Graphs
Gromov Hyperbolicity
Infinite Graphs
Lie Groups
Mathematics
Mathematics and Statistics
Number Theory
Parallelograms
Planar Graphs
Probability Theory and Stochastic Processes
Research Article
Tessellation
Tiles
Topological Groups
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Summary:We prove that under appropriate assumptions adding or removing an infinite amount of edges to a given planar graph preserves its non-hyperbolicity, a result which is shown to be false in general. In particular, we make a conjecture that every tessellation graph of ℝ 2 with convex tiles is non-hyperbolic; it is shown that in order to prove this conjecture it suffices to consider tessellation graphs of ℝ 2 such that every tile is a triangle and a partial answer to this question is given. A weaker version of this conjecture stating that every tessellation graph of ℝ 2 with rectangular tiles is non-hyperbolic is given and partially answered. If this conjecture were true, many tessellation graphs of ℝ 2 with tiles which are parallelograms would be non-hyperbolic.
ISSN:1895-1074
2391-5455
1644-3616
2391-5455
DOI:10.2478/s11533-013-0286-9