Supersolvability and Freeness for $\psi$-graphical Arrangements

Let $G$ be a simple graph on the vertex set $\{v_1,\dots,v_n\}$ with edge set $E$. Let $K$ be a field. The graphical arrangement $\mathcal{A}_G$ in $K^n$ is the arrangement $x_i-x_j=0, v_iv_j \in E$. An arrangement $\mathcal{A}$ is supersolvable if the intersection lattice $L(c(\mathcal{A}))$ of the cone $c(\mathcal{A})$ contains a maximal chain of modular elements. The second author has shown that a graphical arrangement $\mathcal{A}_G$ is supersolvable if and only if $G$ is a chordal graph. He later considered a generalization of graphical arrangements which are called $\psi$-graphical arrangements. He conjectured a characterization of the supersolvability and freeness (in the sense of Terao) of a $\psi$-graphical arrangement. We provide a proof of the first conjecture and state some conditions on free $\psi$-graphical arrangements.