  Motivated by some computations of Feynman integrals and certain conjectures
on mixed Tate motives, Bejleri and Marcolli posed questions about the
$\mathbb{F}_1$-structure (in the sense of torification) on the complement of a
hyperplane arrangement, especially for an arrangement defined in the space of
cycles of a graph. In this paper, we prove that an arrangement has an
$\mathbb{F}_1$-structure if and only if it is Boolean. We also prove that the
arrangement in the cycle space of a graph is Boolean if and only if the cycle
space has a basis consisting of cycles such that any two of them do not share
edges.
