Nice Partitions, Supersolvability, and Freeness in Deformations of Graphic Arrangements

We study affine deformations $\mathcal{A}(G_{\mathcal{S}})$ of graphic arrangements and their cones. Here $G=([n],E(G))$ is a simple graph, $\mathcal{S}=(S_{ij})$ is a family of finite gain sets, and $\mathcal{A}(G_{\mathcal{S}})$ consists of the hyperplanes $x_i-x_j=a$ with $a\in S_{ij}$. We call $