On the connectivity of the disjointness graph of segments of point sets in general position in the plane

TL;DR

This paper investigates the connectivity of the disjointness graph formed by segments between points in general position, establishing a tight lower bound that depends on the number of points.

Contribution

It provides a precise lower bound on the connectivity of the disjointness graph of segments for any point set in general position, and proves this bound is tight.

Findings

Connectivity bound depends on point set size

Bound is tight for all n ≥ 3

Provides combinatorial insights into geometric graphs

Abstract

Let $P$ be a set of $n\geq 3$ points in general position in the plane. The edge disjointness graph $D(P)$ of $P$ is the graph whose vertices are all the closed straight line segments with endpoints in $P$, two of which are adjacent in $D(P)$ if and only if they are disjoint. We show that the connectivity of $D(P)$ is at least $\binom{\lfloor\frac{n-2}{2}\rfloor}{2}+\binom{\lceil\frac{n-2}{2}\rceil}{2}$, and that this bound is tight for each $n\geq 3$.