Planarity is (almost) locally checkable in constant-time

06/21/2020
by   Gábor Elek, et al.
0

Locally checkable proofs for graph properties were introduced by Göös and Suomela <cit.>. Roughly speaking, a graph property is locally checkable in constant-time, if the vertices of a graph having the property can be convinced, in a short period of time not depending on the size of the graph, that they are indeed vertices of a graph having the given property. For a given >0, we call a property -locally checkable in constant-time if the vertices of a graph having the given property can be convinced at least that they are in a graph -close to the given property. We say that a property is almost locally checkable in constant-time, if for all >0, is -locally checkable in constant-time. It is not hard to see that in the universe of bounded degree graphs planarity is not locally checkable in constant-time. However, the main result of this paper is that planarity of bounded degree graphs is almost locally checkable in constant-time. The proof is based on the surprising fact that although graphs cannot be convinced by their planarity or hyperfiniteness, planar graphs can be convinced by their own hyperfiniteness. The reason behind this fact is that the class of planar graphs are not only hyperfinite but possesses Property A of Yu.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
11/07/2018

Every Testable (Infinite) Property of Bounded-Degree Graphs Contains an Infinite Hyperfinite Subproperty

One of the most fundamental questions in graph property testing is to ch...
research
10/05/2021

Local certification of MSO properties for bounded treedepth graphs

The graph model checking problem consists in testing whether an input gr...
research
07/05/2020

Elimination distance to bounded degree on planar graphs

We study the graph parameter elimination distance to bounded degree, whi...
research
08/27/2019

Learning Very Large Graphs with Unknown Vertex Distributions

Recently, Goldreich introduced the notion of property testing of bounded...
research
08/27/2019

Learning Very Large Graphs and Nonsingular Actions of Discrete Groups

Recently, Goldreich introduced the notion of property testing of bounded...
research
08/02/2022

Couboundary Expansion of Sheaves on Graphs and Weighted Mixing Lemmas

We study the coboundary expansion of graphs, but instead of using 𝔽_2 as...
research
03/10/2022

Constructible Graphs and Pursuit

A (finite or infinite) graph is called constructible if it may be obtain...

Please sign up or login with your details

Forgot password? Click here to reset