Before revision:  These wires define homotopy classes with respect to the vertices of G (for the combinatorial embedding of G shared by \Gamma_I and \Gamma_O). These homotopy classes can be represented by orthogonal polylines in \Gamma_I . We argue that the structural difference between the two drawings can be captured by the spirality of the wires in \Gamma_I, which guides our morph from \Gamma_I to \Gamma_O.
After revision:  We can find corresponding wires in \Gamma_I . We argue that the structural difference between the two drawings can be captured by the spirality of the wires in \Gamma_I, which guides our morph from \Gamma_I to \Gamma_O.