Relation-induced connectedness in the digital plane

Abstract

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 \({\mathbb {Z}}^2\) 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.