Bridging graph-theoretical and topological approaches: Connectivity and Jordan curves in the digital plane

Abstract

This article explores the connections between graph-theoretical and topological approaches in the study of the Jordan curve theorem for grids. Building on the foundational work of Rosenfeld, who developed adjacency-based concepts on $Z^2$, and the subsequent introduction of the topological digital plane $K^2$ with the Khalimsky topology by Khalimsky, Kopperman, and Meyer, we investigate the interplay between these perspectives. Inspired by the work of Khalimsky, Kopperman, and Meyer, we define an operator ${\Gamma }^{⁎}$ transforming subsets of $Z^2$ into subsets of $K^2$. This operator is essential for demonstrating how 8-paths, 4-connectivity, and other discrete structures in $Z^2$ correspond to topological properties in $K^2$. Moreover, we address whether the topological Jordan curve theorem for $K^2$ can be derived from the graph-theoretical version on $Z^2$. Our results illustrate the deep and intricate relationship between these two methodologies, shedding light on their complementary roles in digital topology.