On connected subgraph arrangements

TL;DR

This paper studies a special class of hyperplane arrangements derived from graphs, proving new properties and characterizing aspherical arrangements, especially their freeness and exceptions related to the underlying graph structure.

Contribution

It strengthens existing results and introduces new theorems about the freeness and asphericity of connected subgraph arrangements, with a focus on the underlying graph's structure.

Findings

Aspherical connected subgraph arrangements are mostly free.

The only exception is when the underlying graph is the complete graph on 4 nodes.

Freeness is characterized by specific graph properties.

Abstract

Recently, Cuntz and K\"uhne introduced a particular class of hyperplane arrangements stemming from a given graph $G$, so called connected subgraph arrangements $A_G$. In this note we strengthen some of the result from their work and prove new ones for members of this class. For instance, we show that aspherical members withing this class stem from a rather restricted set of graphs. Specifically, if $A_G$ is an aspherical connected subgraph arrangement, then $A_G$ is free with the unique possible exception when the underlying graph $G$ is the complete graph on $4$ nodes.