  We define an admissible decomposition of a graph $E$ into subgraphs $F_1$ and
$F_2$, and consider the intersection graph $F_1\cap F_2$ as a subgraph of both
$F_1$ and $F_2$. We prove that, if the graph $E$ is row finite and its
decomposition into the subgraphs $F_1$ and $F_2$ is admissible, then the graph
C*-algebra $C^*(E)$ of $E$ is the pullback C*-algebra of the canonical
surjections from $C^*(F_1)$ and $C^*(F_2)$ onto $C^*(F_1\cap F_2)$.
