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

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A sign pattern matrix is a matrix whose entries are from the set {+,-,0}. If A is an m× n sign pattern matrix, the qualitative class of A, denoted Q(A), is the set of all real m× n matrices B=[bᵢ,ⱼ] with bᵢ,ⱼ positive (respectively, negative, zero) if aᵢ,ⱼ is + (respectively, -, 0). The minimum rank of a sign pattern matrix A, denoted (A), is the minimum of the ranks of the real matrices in 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.