intertwining argument give the desired result.

One of our motivations for studying this class of $C^{*}$-algebras is that this class contains the class of graph $C^{*}$-algebras whose primitive ideal space is $\widetilde{{\ensuremath{\mathbb{N}}\xspace}}$. In fact, it was shown by the first named author in [@jg:t1graph] that a graph $C^\ast$-algebra with a $T_{1}$ (in particular Hausdorff) primitive ideal space has a canonical $C^{*}$-algebra over $\widetilde{{\ensuremath{\mathbb{N}}\xspace}}$ structure. In this paper, we classify those for which this structure is tight over $\widetilde{{\ensuremath{\mathbb{N}}\xspace}}$ (see Definition \[d:Xalgs\]). This paper contributes to the on going program to classify real rank zero graph $C^{*}$-algebras using ideal-related $K$-theory. See [@ERR:gjogv] for an overview of the classification program for graph $C^{*}$-algebras with finitely many ideals. To the authors knowledge, all known classification results for graph $C^{*}$-algebras involve graph $C^{*}$-algebras with finitely many gauge invariant ideals. Thus, Theorem \[t:classgraph\] is the first classification result for graph $C^{*}$-algebras with infinitely many gauge-invariant ideals of mixed type.

Preliminaries
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In this section, we recall the definition of $C^{*}$-algebras over $X$ and ideal-related $K$-theory (with coefficients) for $C^{*}$-algebras over a totally disconnected space $X$. We also prove several structural properties of $C^{*}$-algebras over $\widetilde{{\ensuremath{\mathbb{N}}\xspace}}$ that will be used throughout the paper.

Throughout the paper, 