Classification of tight $C^{*}$-algebras over the one-point compactification of $\mathbb{N}$

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.