October 16, 2025Open Access

On connected subgraph arrangements

Key Points

Aspherical connected subgraph arrangements show specific characteristics, with most derived from limited graph types.
The connected subgraph arrangements introduced by Cuntz and Kühne have been further examined and refined.
The work discusses the unique conditions under which aspherical members exist, notably related to the complete graph on 4 nodes.
Findings may have implications for understanding hyperplane arrangements in graph theory and their classifications.

Abstract

Recently, Cuntz and K\"uhne introduced a particular class of hyperplane arrangements stemming from a given graph G, so called connected subgraph arrangements AG. 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 AG is an aspherical connected subgraph arrangement, then AG is free with the unique possible exception when the underlying graph G is the complete graph on 4 nodes.

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