  This paper presents a bridge between the theories of wonderful models
associated with toric arrangements and wonderful models associated with
hyperplane arrangements. In a previous work, the same authors noticed that the
model of the toric arrangement of type $A_{n-1}$ is isomorphic to the one of
the hyperplane arrangement of type $A_{n}$; it is natural to ask if there exist
similar isomorphisms between other families of arrangements. The aim of this
paper is to study one such family, namely the family of arrangements defined by
graphs. The main result states that there is indeed an isomorphism between the
model of a toric arrangement defined by a graph $\Gamma$ and the model of a
hyperplane arrangement defined by the cone of $\Gamma$, provided that a
suitable building set is chosen.
