A bad example for the minimum feedback vertex set problem

Let A be a fully indecomposable n x n (0,1) matrix with S(A) nonzero entries. Let M be a minimum feedback vertex set of some strongly connected digraph corresponding to a permuted form of A with nonzero diagonal. Them M has between 1 and S(A)-2n elements. We exhibit a family of matrices, essentially unique, for which any member has permuted forms realizing each of these extremes. The significance for solving sparse linear systems is discussed.