On the Connectedness and Diameter of a Geometric Johnson Graph

TL;DR

This paper studies the connectivity and diameter bounds of a generalized geometric Johnson graph formed by islands of a point set in the plane, revealing conditions for connectivity and estimating its size.

Contribution

It introduces a new class of geometric Johnson graphs based on islands and provides bounds on their diameter and conditions for connectivity.

Findings

Graph is connected for large enough n

Upper and lower bounds on the diameter are established

Connectivity depends on parameters n, k, l

Abstract

Let $P$ be a set of $n$ points in general position in the plane. A subset $I$ of $P$ is called an \emph{island} if there exists a convex set $C$ such that $I = P \cap C$. In this paper we define the \emph{generalized island Johnson graph} of $P$ as the graph whose vertex consists of all islands of $P$ of cardinality $k$, two of which are adjacent if their intersection consists of exactly $l$ elements. We show that for large enough values of $n$, this graph is connected, and give upper and lower bounds on its diameter.