Finitary 1-Simply Connected Digital Spaces

Abstract

Finitary 1-simply connected digital spaces are discrete analogs of the important simply connected spaces in classical topology (i.e., connected spaces in which every loop can be continuously pulled to a point without leaving the space). Loosely speaking, 1-simply connected digital spaces are graphs in which there are no holes larger than a triangle. Many spaces previously studied in digital topology and geometry are instances of this concept. Boundaries in pictures defined over finitary 1-simply connected digital spaces have some desirable general properties; for example, they partition the space into a connected interior and a connected exterior. There is a “one-size-fits-all” algorithm which, given a picture over a finitary 1-simply connected digital space and a boundary face, will return the set of all faces in that boundary, provided only that this set is finite; the proof of correctness of this algorithm is an immediate consequence of the general properties of such spaces.