  A square matrix $A$ is completely positive if $A=BB^T$, where $B$ is a (not
necessarily square) nonnegative matrix. In general, a completely positive
matrix may have many, even infinitely many, such CP factorizations. But in some
cases a unique CP factorization exists. We prove a simple necessary and
sufficient condition for a completely positive matrix whose graph is triangle
free to have a unique CP factorization. This implies uniqueness of the CP
factorization for some other matrices on the boundary of the cone
$\mathcal{CP}_n$ of $n\times n$ completely positive matrices. We also describe
the minimal face of $\mathcal{CP}_n$ containing a completely positive $A$. If
$A$ has a unique CP factorization, this face is polyhedral.
