On connected subgraph arrangements

Abstract In (Cuntz and Kühne 2025), Cuntz and Kühne introduced a particular class of hyperplane arrangements stemming from a given graph, so called connected subgraph arrangements . In this note we strengthen some of the result from (Cuntz and Kühne 2025) and prove new ones for members of this class. For instance, we show that aspherical members within this class stem from a rather restricted set of graphs. Specifically, if $${\mathscr {A}}_G$$ A G is an aspherical connected subgraph arrangement, then $${\mathscr {A}}_G$$ A G is free with the unique possible exception when the underlying graph G is the complete graph on 4 nodes.