  Suppose $G$ is a free product $G = A_1 * A_2* \cdots * A_k * F_N$, where each
of the groups $A_i$ is torsion-free and $F_N$ is a free group of rank $N$. Let
$\mathcal{O}$ be the deformation space associated to this free product
decomposition. We show that the diameter of the projection of the subset of
$\mathcal{O}$ where a given element has bounded length to the
$\mathcal{Z}$-factor graph is bounded, where the diameter bound depends only on
the length bound. This relies on an analysis of the boundary of $G$ as a
hyperbolic group relative to the collection of subgroups $A_i$ together with a
given non-peripheral cyclic subgroup. The main theorem is new even in the case
that $G = F_N$, in which case $\mathcal{O}$ is the Culler-Vogtmann outer space.
In a future paper, we will apply this theorem to study the geometry of free
group extensions.
