Improved upper and lower bounds on the feedback vertex numbers of grids and butterflies

We improve upon the best known upper and lower bounds on the sizes of minimal feedback vertex sets in butterflies. Also, we construct new feedback vertex sets in grids so that for a large number of pairs (n,m)(n,m), the size of our feedback vertex set in the grid Mn,mMn,m matches the best known lower bound, and for all other pairs it differs from this lower bound by at most 2.