  The Constrained Minimum Determinant Factor Analysis (CMDFA) setting was
motivated by Wyner's common information problem where we seek a latent
representation of a given Gaussian vector distribution with the minimum mutual
information under certain generative constraints. In this paper, we explore the
algebraic structures of the solution space of the CMDFA, when the underlying
covariance matrix $\Sigma_x$ has an additional latent graphical constraint,
namely, a latent star topology. In particular, sufficient and necessary
conditions in terms of the relationships between edge weights of the star graph
have been found. Under such conditions and constraints, we have shown that the
CMDFA problem has either a rank one solution or a rank $n-1$ solution where $n$
is the dimension of the observable vector. Further results are given in regards
to the solution to the CMDFA with $n-1$ latent factors.
