Bounds on the base of primitive nearly reducible sign pattern matrices

In [J.Y. Shao, L.H. You, Bound on the base of irreducible generalized sign pattern matrices, Discrete Math., in press], Shao and You extended the concept of the base from powerful sign pattern matrices to non-powerful (and generalized) sign pattern matrices. In this paper, we study the base for primitive non-powerful nearly reducible sign pattern (and generalized sign pattern) matrices. We obtain sharp upper bounds, together with complete characterization of the equality cases of the base for primitive nearly reducible sign pattern (and generalized sign pattern) matrices. We also show that there exist “gaps” in the base set of the classes of such matrices.