 
Summary: Proximity Graphs inside Large Weighted Graphs
Bernardo M. ŽAbrego
Ruy FabilaMonroy
Silvia FernŽandezMerchant
David FloresPe~naloza
Ferran Hurtado§
Henk Meijer¶
Vera SacristŽan§
Maria Saumell§
Abstract
Given a large weighted graph G = (V, E) and a subset U of V , we define several graphs
with vertex set U in which two vertices are adjacent if they satisfy some prescribed proximity
rule. These rules use the shortest path distance in G and generalize the proximity rules that
generate some of the most common proximity graphs in Euclidean spaces. We prove basic
properties of the defined graphs and provide algorithms for their computation.
Keywords Neighborhood; proximity graphs; Voronoi diagrams; weighted graphs.
1 Introduction
A basic need in spatial data analysis, from statistics to pattern recognition and wherever shape
extraction is involved, is to decide closeness and neighborhoods among elements of a given input.
When the data are described as points in Euclidean spaces, proximity graphs in which two of
