Non-surjective pullbacks of graph C*-algebras from non-injective pushouts of graphs

We find a substantial class of pairs of $\ast$-homomorphisms between graph C*-algebras of the form $C^{\ast }(E)\hookrightarrow C^{\ast }(G)\twoheadleftarrow C^{\ast }(F)$ whose pullback C*-algebra is an AF graph C*-algebra. Our result can be interpreted as a recipe for determining the quantum space obtained by shrinking a quantum subspace. There is a variety of examples from noncommutative topology, such as quantum complex projective spaces (including the standard Podle's quantum sphere) or quantum teardrops, that instantiate the result. Furthermore, to go beyond AF graph C*-algebras, we consider extensions of graphs over sinks and prove an analogous theorem for the thus obtained graph C*-algebras.