Digital Jordan curves and surfaces with respect to a graph connectedness

Abstract

We introduce a graph connectedness induced by a given set of paths of the same length. We focus on the 2-adjacency graph on the digital line ℤ with a certain set of paths of length n for every positive integer n. The connectedness in the strong product of two and three copies of the graph is used to define digital Jordan curves and digital Jordan surfaces, respectively. Such definitions build on an edge-to-edge tiling with triangles in the digital plane and a face-to-face tiling by cubes, prisms and pyramids in the (3D) digital space, respectively.