We study two classical connectivity-preserving parallel shrinking algorithms proposed to recognize and label two-dimensional connected components of binary images. The algorithms we consider were developed by Beyer [Recognition of topological invariants by iterative arrays, Ph.D. Thesis, MIT, 1969, p. 144] and Levialdi [Commun. ACM 15 (1) (1972) 7] independently for the purpose of shrinking 4-connected and 8-connected components of binary images in linear time, respectively. It is shown that those two independently developed algorithms are closely related and in a sense they are in a dual relation such that, for any initially given binary image and its inverted one, one algorithm produces, simultaneously, an image which is dual of the one produced by the other, step-by-step.