Relative graphs and pullbacks of relative Toeplitz graph algebras

In this note we generalize a result from a recent paper of Hajac, Reznikoff and Tobolski (2020). In that paper they give conditions they call admissibility on a pushout diagram in the category of directed graphs implying that the $C^*$-algebras of the graphs form a pullback diagram. We consider a larger category of relative graphs that correspond to relative Toeplitz graph algebras. In this setting we give necessary and sufficient conditions on the pushout to get a pullback of $C^*$-algebras.