Edge complexity of geometric graphs on convex independent point sets

TL;DR

This paper investigates the edge complexity of Locally Gabriel graphs and Unit distance graphs on convex independent point sets, providing simpler proofs and improved bounds on their number of edges.

Contribution

It offers a simpler proof for the edge bound of LGGs and establishes a tighter linear bound for UDGs on convex independent point sets.

Findings

LGGs on convex independent point sets have at most 2n log n + O(n) edges

UDGs on convex independent point sets have O(n) edges, improving previous bounds

The paper simplifies existing proofs and tightens bounds on graph edge complexity

Abstract

In this paper, we focus on a generalised version of Gabriel graphs known as Locally Gabriel graphs ($LGGs$) and Unit distance graphs ($UDGs$) on convexly independent point sets. $UDGs$ are sub graphs of $LGGs$. We give a simpler proof for the claim that $LGGs$ on convex independent point sets have $2n \log n + O(n)$ edges. Then we prove that unit distance graphs on convex independent point sets have $O(n)$ edges improving the previous known bound of $O(n \log n)$.