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On the treewidth of random geometric graphs and percolated grids

Part of: Geometric probability and stochastic geometry Graph theory Equilibrium statistical mechanics

Published online by Cambridge University Press:  17 March 2017

Anshui Li  and
Tobias Müller
Anshui Li*
Affiliation:
Hangzhou Normal University
Tobias Müller*
Affiliation:
Utrecht University
*
* Postal address: Department of Mathematics, Hangzhou Normal University, Hangzhou, 310000, P.R. China. Email address: anshuili@hznu.edu.cn
** Postal address: Mathematical Institute, Utrecht University, Utrecht, 3508TA, The Netherlands. Email address: t.muller@uu.nl
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Abstract

In this paper we study the treewidth of the random geometric graph, obtained by dropping n points onto the square [0,√n]2 and connecting pairs of points by an edge if their distance is at most r=r(n). We prove a conjecture of Mitsche and Perarnau (2014) stating that, with probability going to 1 as n→∞, the treewidth of the random geometric graph is 𝜣(r√n) when lim inf r>rc, where rc is the critical radius for the appearance of the giant component. The proof makes use of a comparison to standard bond percolation and with a little bit of extra work we are also able to show that, with probability tending to 1 as k→∞, the treewidth of the graph we obtain by retaining each edge of the k×k grid with probability p is 𝜣(k) if p>½ and 𝜣(√log k) if p<½.

Keywords

Random geometric graphtreewidth

MSC classification

Primary: 05C80: Random graphs
Secondary: 60D05: Geometric probability and stochastic geometry 82B43: Percolation
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 1 , March 2017 , pp. 49 - 60
Copyright
Copyright © Applied Probability Trust 2017 

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