On connected subgraph arrangements

Abstract

Recently, Cuntz and Kühne 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.