Joseph Rogge | Searcharxiv

arXiv2021

On Clusters that are Separated but Large

$\renewcommand{\Re}{\mathbb{R}}$Given a set $P$ of $n$ points in $\Re^d$, consider the problem of computing $k$ subsets of $P$ that form clusters that are well-separated from each other, and each of them is large (cardinality wise). We provide tight upper and lower bounds, and corresponding algorithms, on the quality of separation, and the size of the clusters that can be computed, as a function of $n,d,k,s$, and $Φ$, where $s$ is the desired separation, and $Φ$ is the spread of the point set $P$.

cs.CG↗

arXiv2020

Convex geometries representable by at most 5 circles on the plane

A convex geometry is a closure system satisfying the anti-exchange property. In this work we document all convex geometries on 4- and 5-element base sets with respect to their representation by circles on the plane. All 34 non-isomorphic geometries on a 4-element set can be represented by circles, and of the 672 geometries on a 5-element set, we made representations of 623. Of the 49 remaining geometries on a 5-element set, one was already shown not to be representable due to the Weak Carousel property, as articulated by Adaricheva and Bolat (Discrete Mathematics, 2019). In this paper we show that 7 more of these convex geometries cannot be represented by circles on the plane, due to what we term the Triangle Property.

math.CO↗