For digital images, there is an established homotopy equivalence relation which parallels that of classical topology. Many classical homotopy equivalence invariants, such as the Euler characteristic and the homology groups, do not remain invariants in the digital setting. This paper develops a numerical digital homotopy invariant and begins to catalog all possible connected digital images on a small number of points, up to homotopy equivalence.

We continue the work of [4,2,3], in which we discuss published assertions that are incorrect or incorrectly proven; that are severely limited or reduce to triviality; or that we improve upon.MSC: 54H25

We continue the study of freezing sets in digital topology, introduced in [2]. We show how to find a minimal freezing set for a "thick" convex disk X in the digital plane Z 2 . We give examples showing the significance of the assumption that X is convex.We use Z to indicate the set of integers and R for the set of real numbers. For a finite set X, we denote by #X the number of distinct members of X.

We study connectivity preserving multivalued functions [10] between digital images. This notion generalizes that of continuous multivalued functions [6,7] studied mostly in the setting of the digital plane Z 2 . We show that connectivity preserving multivalued functions, like continuous multivalued functions, are appropriate models for digital morpholological operations. Connectivity preservation, unlike continuity, is preserved by compositions, and generalizes easily to higher dimensions and arbitrary adjacency relations.

The current paper focuses on fundamental groups and Euler characteristics of various digital models of the 2-dimensional sphere. For all models that we consider, we show that the fundamental groups are trivial, and compute the Euler characteristics (which are not always equal). We consider the connected sum of digital surfaces and investigate how this operation relates to the fundamental group and Euler characteristic. We also consider two related but different notions of a digital image having "no holes," and relate this to the triviality of the fundamental group. Many of our results have origins in the paper [15] by S.-E. Han, which contains many errors. We correct these errors when possible, and leave some open questions. We also present some original results.2010 MSC: 68R10; 55Q40.