  We prove a strong classification result for a certain class of
$C^{*}$-algebras with primitive ideal space $\widetilde{\mathbb{N}}$, where
$\widetilde{\mathbb{N}}$ is the one-point compactification of $\mathbb{N}$.
This class contains the class of graph $C^{*}$-algebras with primitive ideal
space $\widetilde{\mathbb{N}}$. Along the way, we prove a universal coefficient
theorem with ideal-related $K$-theory for $C^{*}$-algebras over
$\widetilde{\mathbb{N}}$ whose $\infty$ fiber has torsion-free $K$-theory.
