Adjacency in Digital Pictures

Let S be a subset of a digital picture, and let s be the complement of S. It is well known that the connected components of S and s, under the relation “is adjacent to,≓ form a tree, and algorithms for constructing this tree have been devised. The main purpose of this paper is to prove that the components do form a tree, and in the process, to provide a basis for proving the validity of the tree-constructing algorithms.