Completely positive house matrices

In this paper, we discuss the complete positivity of house matrices, i.e., doubly nonnegative matrices whose graph is a cycle (1,2,…,n) with a chord between vertices 1 and 3. We describe the set of all possible supports of the columns of a nonnegative matrix B,such thatA=BBT; show that the cp-rank of a completely positive house matrix is at least n-1; give a complete characterization of the singular completely positive house matrices; show that their cp-rank is n-1 or n and characterize each case.