Mixed dominating matrices

We characterize the class of matrices for which the set of supports of nonnegative vectors in the null space can be determined by the signs of the entries of the matrix. This characterization is in terms of mixed dominating matrices, which are defined by the nonexistence of square submatrices that have nonzeros of opposite sign in each row. The class of mixed dominating matrices is contained in the class of L-matrices from the theory of sign-solvability, and generalizes the class of S-matrices. We give a polynomial-time algorithm to decide if a matrix is mixed dominating. We derive combinatorial conditions on the face lattice of a Gale transform of a matrix in this class.