Disjoint edges in geometric graphs

TL;DR

This paper investigates properties of geometric graphs, establishing lower bounds on disjoint edge pairs under certain conditions and providing upper bounds on the total number of edges based on disjointness constraints.

Contribution

It introduces new bounds on disjoint edges in geometric graphs and characterizes the maximum number of edges given disjointness limitations, with proofs of tightness.

Findings

At least rac{n}{2}inom{2e/n}{3} disjoint edge pairs exist under specified conditions.

Maximum edges are bounded by rac{n(1+8m)^{1/2}+3}{4} when each edge disjoint from at most m edges.

Abstract

A geometric graph is a graph drawn in the plane so that its vertices and edges are represented by points in general position and straight line segments, respectively. A vertex of a geometric graph is called pointed if it lies outside of the convex hull of its neighbours. We show that for a geometric graph with $n$ vertices and $e$ edges there are at least $\frac{n}{2}\binom{2e/n}{3}$ pairs of disjoint edges provided that $2e\geq n$ and all the vertices of the graph are pointed. Besides, we prove that if any edge of a geometric graph with $n$ vertices is disjoint from at most $ m $ edges, then the number of edges of this graph does not exceed $n(\sqrt{1+8m}+3)/4$ provided that $n$ is sufficiently large. These two results are tight for an infinite family of graphs.