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On the relation between graph distance and Euclidean distance in random geometric graphs

Part of: Graph theory Discrete mathematics in relation to computer science

Published online by Cambridge University Press:  19 September 2016

J. Díaz ,
D. Mitsche ,
G. Perarnau  and
X. Pérez-Giménez
J. Díaz*
Affiliation:
Universitat Politècnica de Catalunya and BGSMath
D. Mitsche*
Affiliation:
Université Nice Sophia Antipolis
G. Perarnau*
Affiliation:
Universitat Politècnica de Catalunya
X. Pérez-Giménez*
Affiliation:
University of Waterloo
*
* Postal address: Department of Computer Science, Universitat Politècnica de Catalunya, Jordi Girona Salgado 1, 08034 Barcelona, Spain. Email address: diaz@lsi.upc.edu
** Postal address: Laboratoire J. A. Dieudonné, Université Nice Sophia Antipolis, Parc Valrose, 06108 Nice cedex 02, France. Email address: dmitsche@unice.fr
*** Postal address: Department de Matemàtica Aplicada IV, Universitat Politècnica de Catalunya, Jordi Girona Salgado 1, 08034 Barcelona, Spain. Email address: guillem.perarnau@ma4.upc.edu
**** Postal address: Department of Combinatorics and Optimization, University of Waterloo, Waterloo ON, N2L 3G1, Canada. Email address: xperez@uwaterloo.ca
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Abstract

Given any two vertices u, v of a random geometric graph G(n, r), denote by dE(u, v) their Euclidean distance and by dE(u, v) their graph distance. The problem of finding upper bounds on dG(u, v) conditional on dE(u, v) that hold asymptotically almost surely has received quite a bit of attention in the literature. In this paper we improve the known upper bounds for values of r=ω(√logn) (that is, for r above the connectivity threshold). Our result also improves the best known estimates on the diameter of random geometric graphs. We also provide a lower bound on dE(u, v) conditional on dE(u, v).

Keywords

Random geometric graphgraph distanceEuclidean distancediameter

MSC classification

Primary: 05C80: Random graphs
Secondary: 68R10: Graph theory (including graph drawing)
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 3 , September 2016 , pp. 848 - 864
Copyright
Copyright © Applied Probability Trust 2016 

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