The minimum rank of a sign pattern matrix with a 1-separation

A sign pattern matrix is a matrix whose entries are from the set {+,−,0}{+,−,0}. If A   is an m×nm×n sign pattern matrix, the qualitative class of A  , denoted Q(A)Q(A), is the set of all real m×nm×n matrices B=[bi,j]B=[bi,j] with bi,jbi,j positive (respectively, negative, zero) if ai,jai,j is + (respectively, −, 0). The minimum rank of a sign pattern matrix A  , denoted mr(A)mr(A), is the minimum of the ranks of the real matrices in Q(A)Q(A). Determination of the minimum rank of a sign pattern matrix is a longstanding open problem.For the case that the sign pattern matrix has a 1-separation, we present a formula to compute the minimum rank of a sign pattern matrix using the minimum ranks of certain generalized sign pattern matrices associated with the 1-separation.