---
title: Sparse geometric graphs with small dilation
url: https://www.emergentmind.com/papers/0702080
type: paper
arxiv_id: '0702080'
arxiv_url: https://arxiv.org/abs/0702080
published: '2007-02-14'
categories:
- cs.CG
---

# Sparse geometric graphs with small dilation

## Abstract

Given a set S of n points in R^D, and an integer k such that 0 <= k < n, we show that a geometric graph with vertex set S, at most n - 1 + k edges, maximum degree five, and dilation O(n / (k+1)) can be computed in time O(n log n). For any k, we also construct planar n-point sets for which any geometric graph with n-1+k edges has dilation Omega(n/(k+1)); a slightly weaker statement holds if the points of S are required to be in convex position.