On the number of CP factorizations of a completely positive matrix

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.