  Let $S$ be an upper cluster algebra, which is a subalgebra of $R$. Suppose
that there is some cluster variable $x_e$ such that ${R}_{{x}_e} = S[{x}_e^{\pm
1}]$. We try to understand under which conditions ${R}$ is an upper cluster
algebra, and how the quiver of $R$ relates to that of $S$. Moreover, if the
restriction of $(\Delta,W)$ to some subquiver is a cluster model, we give a
sufficient condition for $(\Delta,W)$ itself being a cluster model. As an
application, we show that the semi-invariant ring of any complete $m$-tuple
flags is an upper cluster algebra whose quiver is explicitly given. Moreover,
the quiver with its rigid potential is a polyhedral cluster model.
