The graph $T_n$ on n vertices, consisiting of n-2 triangles with common base, is completely positive.
Let $A$ be a matrix realization of $T_n$. We may assume that $A$ is nonsingular and that its diagonal entries are all equal to $1$. If the common base $T_n$ is $\{1,2\}$, then $A$ is of form
$$A= \begin{pmatrix} B &C \\ C^T & I_{n-2}\end{pmatrix}$$
where $B \in Mat_{2 \times 2}$ nonegative matrix, and $C \in Mat_{\{2 \times (n-2)\}}$ is a positive matrix.
So $$A= \begin{pmatrix} \frac{A}{I_{n-2}} &0 \\ 0& 0_{n-2} \end{pmatrix}+ \begin{pmatrix} C\\ I_{n-2} \end{pmatrix} \begin{pmatrix} C^T I_{n-2}\end{pmatrix}$$ Then we end the proof with Schur complement and Theorem about sum of certain matrix. Is there any easier way to prove that?