I will explain how the graph C*-algebras of a trimmable graph can be decomposed as U(1)-equivariant pullback of two simpler C*-algebras. A main example is given by the algebra of Vaksman-Soibelman quantum sphere, that can be realized as pushout of a lower dimensional quantum sphere and the product of a quantum ball with a circle (we understand the pullback of C*-algebras as pushout of the underlying "noncommutative spaces"). The U(1)-invariant part of this pullback diagram gives a "CW complex" realization of quantum projective spaces that allows to give an explicit description of the K-theory generators. Further examples include quantum lens spaces, one-loop extensions of Cuntz algebras of the Toeplitz algebra (joint work with Francesca Arici, Piotr M. Hajac and Mariusz Tobolski). I will then illustrate an alternative cell complex decomposition of quantum projective spaces in terms of "non-spherical" noncommutative balls (products of quantum disks) attached along multipullback quantum spheres (joint work with P.M. Hajac, T. Maszczyk, A. Sheu and B. Zieli\'nski).