We present an algorithm for constructing pairs of an invertible mapping A and an affine mapping B such that AS = SB for a given S-box. For doing we so, we introduce and analyse the link graph of an S-box. We apply the algorithm to the eight DES S-boxes. All obtained pairs (A,B) are those reported in previous work, in which it was required that both A and B are invertible affine mappings. In particular, the relaxation that A need not be affine does not yield new pairs.