Characterizing Nice Partition of Graphical Arrangements

The successive works of Terao as well as Stanley revealed that, for graphical arrangements, supersolvability and the existence of nice partitions are equivalent properties, both characterized by chordal graphs. In this paper, we further prove that every nice partition of a graphical arrangement arises precisely from a maximal modular chain in its intersection lattice. Moreover, we establish two converses to classical results of Orlik and Terao on nice partitions.