  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.
