  We prove that the graph C*-algebra $C^*(E)$ of a trimmable graph $E$ is
$U(1)$-equivariantly isomorphic to a pullback C*-algebra of a subgraph
C*-algebra $C^*(E'')$ and the C*-algebra of functions on a circle tensored with
another subgraph C*-algebra $C^*(E')$. This allows us to unravel the structure
and K-theory of the fixed-point subalgebra $C^*(E)^{U(1)}$ through the
(typically simpler) C*-algebras $C^*(E')$, $C^*(E'')$ and $C^*(E'')^{U(1)}$. As
examples of trimmable graphs, we consider one-loop extensions of the standard
graphs encoding respectively the Cuntz algebra $\mathcal{O}_2$ and the Toeplitz
algebra $\mathcal{T}$. Then we analyze equivariant pullback structures of
trimmable graphs yielding the C*-algebras of the Vaksman-Soibelman quantum
sphere $S^{2n+1}_q$ and the quantum lens space $L_q^3(l; 1,l)$, respectively.
