We give a classification result for a certain class of $C^{*}$-algebras
$\mathfrak{A}$ over a finite topological space $X$ in which there exists an
open set $U$ of $X$ such that $U$ separates the finite and infinite
subquotients of $\mathfrak{A}$. We will apply our results to $C^{*}$-algebras
arising from graphs.