Relative hyperbolicity for automorphisms of free products and free groups

We prove that for a free product G with free factor system g, any automorphism phi preserving g, atoroidal (in a sense relative to g) and none of whose power send two different conjugates of subgroups in g on conjugates of themselves by the same element, gives rise to a semidirect product G (sic)(phi) Z that is relatively hyperbolic with respect to suspensions of groups in g. We recover a theorem of Gautero-Lustig and Ghosh that, if G is a free group, phi an automorphism of G, and g is its family of polynomially growing subgroups, then the semidirect product by phi is relatively hyperbolic with respect to the suspensions of these subgroups. We apply the first result to the conjugacy problem for certain automorphisms (atoroidal and toral) of free products of abelian groups.