Relation-induced connectedness in the digital plane

n-Ary relation, Connectedness, Digital plane, Khalimsky topology, Jordan curve

We introduce and discuss a connectedness induced by n-ary relations (n > 1 an integer) on their underlying sets. In particular, we focus on certain n-ary relations with the induced connectedness allowing for a definition of digital Jordan curves. For every integer n > 1, we introduce one such n-ary relation on the digital plane Z2 and prove a digital analogue of the Jordan curve theorem for the induced connectedness. It follows that these n-ary relations may be used as convenient structures on the digital plane for the study of geometric properties of digital images. For n = 2, such a structure coincides with the (specialization order of the) Khalimsky topology and, for n > 2, it allows for a variety of Jordan curves richer than that provided by the Khalimsky topology.