  Let $\phi$ be an atoroidal outer automorphism of the free group $F_n$. We
study the Gromov boundary of the hyperbolic group $G_{\phi} = F_n
\rtimes_{\phi} \mathbb{Z}$. We explicitly describe a family of embeddings of
the complete bipartite graph $K_{3,3}$ into $\partial G_\phi$. To do so, we
define the directional Whitehead graph and prove that an indecomposable
$F_n$-tree is Levitt type if and only if one of its directional Whitehead
graphs contains more than one edge. As an application, we obtain a direct proof
of Kapovich-Kleiner's theorem that $\partial G_\phi$ is homeomorphic to the
Menger curve if the automorphism is atoroidal and fully irreducible.
